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Big Ideas Math Book Algebra 2 Answer Key Chapter 5 Rational Exponents and Radical Functions

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• Rational Exponents and Radical Functions Maintaining Mathematical Proficiency – Page 235
• Rational Exponents and Radical Functions Mathematical Practices – Page 236
• Lesson 5.1 nth Roots and Rational Exponents – Page(238-242)
• nth Roots and Rational Exponents 5.1 Exercises – Page(241-242)
• Lesson 5.2 Properties of Rational Exponents and Radicals – Page(244-250)
• Properties of Rational Exponents and Radicals 5.2 Exercises – Page(248-250)
• Lesson 5.3 Graphing Radical Functions – Page(252-258)
• Graphing Radical Functions 5.3 Exercises – Page(256-258)
• Rational Exponents and Radical Functions Study Skills – Page 259
• Rational Exponents and Radical Functions 5.1 – 5.3 Quiz – Page 260
• Lesson 5.4 Solving Radical Equations and Inequalities – Page(262-268)
• Solving Radical Equations and Inequalities 5.4 Exercises – Page(266-268)
• Lesson 5.5 Performing Function Operations – Page(270-274)
• Performing Function Operations 5.5 Exercises – Page(273-274)
• Lesson 5.6 Inverse of a Function – Page(276-284)
• Inverse of a Function 5.6 Exercises – Page(281-284)
• Rational Exponents and Radical Functions Performance Task – Page 285
• Rational Exponents and Radical Functions Chapter Review – Page(286-288)
• Rational Exponents and Radical Functions Chapter Test – Page 289
• Rational Exponents and Radical Functions Cumulative Assessment – Page(290-291)

Rational Exponents and Radical Functions Maintaining Mathematical Proficiency

Simplify the expression

Question 1.
y⁶ • y
Answer: y⁷
Explanation:
Given,
y⁶ • y
When bases are equal powers should be added.
a^m . aⁿ = a^m+n
y⁶ • y = y⁶⁺¹ = y⁷

Question 2.
\(\frac{n^{4}}{n^{3}}\)​
Answer: n
Explanation:
Given,
\(\frac{n^{4}}{n^{3}}\)​
a^m / aⁿ = a^m-n
= n^4-3
= n¹
So, \(\frac{n^{4}}{n^{3}}\)​ = n

Question 3.
\(\frac{x^{5}}{x^{6} \cdot x^{2}}\)
Answer: n^-3
Explanation:
Given,

Question 4.
\(\frac{x^{-6}}{x^{5}}\) • 3x²
Answer: 3x^-9
Explanation:
Given,
\(\frac{x^{-6}}{x^{5}}\) • 3x²
a^m / aⁿ = a^m-n
= x^-6-5 .  3x²
= x^-11 .  3x²
= 3x^-9
So, \(\frac{x^{-6}}{x^{5}}\) • 3x² = 3x^-9

Question 5.
\(\left(\frac{4 w^{3}}{2 z^{2}}\right)^{3}\)
Answer: \(\left(\frac{2 w^{3}}{z^{2}}\right)^{3}\)
Explanation:
Given,
\(\left(\frac{4 w^{3}}{2 z^{2}}\right)^{3}\)
4/2 = 2
\(\left(\frac{2 w^{3}}{z^{2}}\right)^{3}\)

Question 6.
\(\left(\frac{m^{7} \cdot m}{z^{2} \cdot m^{3}}\right)^{2}\)
Answer: (m⁵/z²)²
Explanation:
Given,
\(\left(\frac{m^{7} \cdot m}{z^{2} \cdot m^{3}}\right)^{2}\)
= (m⁸/z²m³)²
= (m⁵/z²)²
So, \(\left(\frac{m^{7} \cdot m}{z^{2} \cdot m^{3}}\right)^{2}\) = (m⁵/z²)²

Solve the literal equation for y.

Question 7.
4x + y = 2
Answer: 2 – 4x
Explanation:
Given,
4x + y = 2
Rewrite the equation so a different variable stands alone on one side of the equal sign.
We have to make y the subject.
y = 2 – 4x
So, the equation for y is 2 – 4x.

Question 8.
x − \(\frac{1}{3}\)y = −1
Answer: 3(x + 1)
Explanation:
Given,
x − \(\frac{1}{3}\)y = −1
Rewrite the equation so a different variable stands alone on one side of the equal sign.
Make y as the subject.
− \(\frac{1}{3}\)y = -1 – x
-(\(\frac{1}{3}\)y = -(1 + x)
\(\frac{1}{3}\)y = x + 1
y = 3(x + 1)
So, the equation for y is 3(x + 1).

Question 9.
2y − 9 = 13x
Answer:

Question 10.
2xy + 6y = 10
Answer: 5/3 + x
Explanation:
Given,
2xy+6y=10
Take 2 as the common factor.
2(xy+3y)=2(5)
Divide by 2 on both sides.
2(xy+3y)​/2=2(5)/2
xy+3y=5xy+3y
=5xy+3y=5
Rewrite the equation so a different variable stands alone on one side of the equal sign.
Make y as the subject.
y(x+3)=5
y(x+3)​/x+3 = 5/3+x
y = 5/3 + x
So, the equation for y is 5/3 + x​​​

Question 11.
8x − 4xy = 3
Answer: -3/4x​+2
Explanation:
Given,
8x − 4xy = 3
Apply inverse operations to isolate $y.$
8x−4xy=3
4x(2−y)=3
Rewrite the equation so a different variable stands alone on one side of the equal sign.
Make y as the subject.
2−y=3/4x
–y=3/4x−2
y=–1(3/4x−2)
y=-3/4x​+2
So, the equation for y is -3/4x​+2​

Question 12.
6x + 7xy = 15
Answer: (15 – 6x)/7x
Explanation:
Given,
6x + 7xy = 15
Rewrite the equation so a different variable stands alone on one side of the equal sign.
Make y as the subject.
7xy = 15 – 6x
y = (15 – 6x)/7x
So, the equation for y is (15 – 6x)/7x

Question 13.
ABSTRACT REASONING Is the order in which you apply properties of exponents important? Explain your reasoning.
Answer: Exponents are values that are powers that show how many times we have to multiply the base number by itself. The exponents are important to write the values of numbers in a simplified form.
The properties of the exponent are
1. Product of powers
2. Quotient of powers
3. Power to a power
4. Power of a product
5. Power of a quotient

Rational Exponents and Radical Functions Mathematical Practices

Monitoring Progress

Question 1.
Use the Pythagorean Theorem to find the exact lengths of a, b, c, and d in the figure.

Question 2.
Use a calculator to approximate each length to the nearest tenth of an inch.
Answer:
We can find the length and angle measured to the nearest tenth using the calculator.
a = √2
a = 1.414
a ≈ 1.41
b = √3
b = 1.732
b ≈ 1.73
c = 2
c is the whole number
d = √5
d = 2.2360
d ≈ 2.24

Question 3.
Use a ruler to check the reasonableness of your answers.

Answer:
a = √2
b = √3
c = 2
d = √5
Explanation:
Given,
s = 1 in.
1² + 1² = a²
a = √1 + 1
a = √2
The measure of hypotenuse a is √2.
√2² + 1² = b²
2 + 1 = b²
b² = 3
b = √3
The measure of hypotenuse b is √3.
√3² + 1² = c²
c² = 3 + 1
c² = 4
c = 2
The measure of hypotenuse c is 2.
2² + 1² = d²
d² = 4 + 1
d² = 5
d = √5
The measure of hypotenuse d is √5.

Lesson 5.1 nth Roots and Rational Exponents

Essential Question

How can you use a rational exponent to represent a power involving a radical?
Previously, you learned that the nth root of a can be represented as
\(\sqrt[n]{a}\) = \(a^{1 / n}\) Definition of rational exponent
for any real number a and integer n greater than 1.

EXPLORATION 1
Exploring the Definition of a Rational Exponent
Work with a partner. Use a calculator to show that each statement is true.
a. \(\sqrt{9}\) = \(9^{1 / 2}\)
Answer:
Given,
\(\sqrt{9}\) = \(9^{1 / 2}\)
In order to rewrite a radical using a fractional exponent, the power to which the radicand becomes the numerator and the root becomes the denominator.
Square is nothing but 2.
So, \(\sqrt{9}\) can be written in the fractional exponent as \(9^{1 / 2}\).
The statement \(\sqrt{9}\) = \(9^{1 / 2}\) is true.

b. \(\sqrt{2}\) = \(2^{1 / 2}\)
Answer:
Given,
\(\sqrt{2}\) = \(2^{1 / 2}\)
In order to rewrite a radical using a fractional exponent, the power to which the radicand becomes the numerator and the root becomes the denominator.
Square is nothing but 2.
So, \(\sqrt{2}\) can be written in the fractional exponent as \(2^{1 / 2}\).
The statement is true.

c. \(\sqrt[3]{8}\) = \(8^{1 / 3}\)
Answer:
Given,
\(\sqrt[3]{8}\) = \(8^{1 / 3}\)
In order to rewrite a radical using a fractional exponent, the power to which the radicand becomes the numerator and the root becomes the denominator.
Cube root is nothing but 1/3.
\(\sqrt[3]{8}\) can be written in the fractional exponent as \(8^{1 / 3}\).
The statement is true.

d. \(\sqrt[3]{3}\) = \(3^{1 / 3}\)
Answer:
Given,
\(\sqrt[3]{3}\) = \(3^{1 / 3}\)
In order to rewrite a radical using a fractional exponent, the power to which the radicand becomes the numerator and the root becomes the denominator.
Cube root is nothing but 1/3.
\(\sqrt[3]{3}\) can be written in the fractional exponent as \(3^{1 / 3}\).
The statement is true.

e. \(\sqrt[4]{16}\) = \(16^{1 / 4}\)
Answer:
Given,
\(\sqrt[4]{16}\) = \(16^{1 / 4}\)
In order to rewrite a radical using a fractional exponent, the power to which the radicand becomes the numerator and the root becomes the denominator.
The fourth root is nothing but 1/4.
\(\sqrt[4]{16}\) can be written in the fractional exponent as \(16^{1 / 4}\).
The statement is true.

f. \(\sqrt[4]{12}\) = \(12^{1 / 4}\)
Answer:
Given,
\(\sqrt[4]{12}\) = \(12^{1 / 4}\)
In order to rewrite a radical using a fractional exponent, the power to which the radicand becomes the numerator and the root becomes the denominator.
The fourth root is nothing but 1/4.
\(\sqrt[4]{12}\) can be written in the fractional exponent as \(12^{1 / 4}\).

EXPLORATION 2
Writing Expressions in Rational Exponent Form

Work with a partner. Use the definition of a rational exponent and the properties of exponents to write each expression as a base with a single rational exponent. Then use a calculator to evaluate each expression. Round your answer to two decimal places.

a. \((\sqrt{5})^{3}\)
Answer: 11.08
Explanation:
Given,
\((\sqrt{5})^{3}\)
The square root of 5 is 2.23
(\(\sqrt{5}\))³ = (2.23)³ = 11.08

b. \((\sqrt[4]{4})^{2}\)
Answer: 1.99
Explanation:
Given,
\((\sqrt[4]{4})^{2}\)
The fourth root of 4 is 1.414
\((\sqrt[4]{4})^{2}\) = (1.414)² = 1.99

c. \((\sqrt[3]{9})^{2}\)
Answer: 4.32
Explanation:
Given,
\((\sqrt[3]{9})^{2}\)
The cube root of 9 is 2.08
\((\sqrt[3]{9})^{2}\) = (2.08)² = 4.32

d. \((\sqrt[5]{10})^{4}\)
Answer: 6.23
Explanation:
Given,
\((\sqrt[5]{10})^{4}\)
The fifth root of 10 is 1.58
\((\sqrt[5]{10})^{4}\) = (1.58)⁴ = 6.23

e. \((\sqrt{15})^{3}\)
Answer: 57.96
Explanation:
Given,
\((\sqrt{15})^{3}\)
the square root of 15 is 3.87
\((\sqrt{15})^{3}\) = (\(\sqrt{15}\))³ = (3.87)³ = 57.96

f. \((\sqrt[3]{27})^{4}\)
Answer: 81
Explanation:
Given,
\((\sqrt[3]{27})^{4}\)
The cube root of 27 is 3.
\((\sqrt[3]{27})^{4}\) = (3)⁴ = 81

EXPLORATION 3
Writing Expressions in Radical Form
Work with a partner. Use the properties of exponents and the definition of a rational exponent to write each expression as a radical raised to an exponent. Then use a calculator to evaluate each expression. Round your answer to two decimal places.
Sample \(5^{2 / 3}\) = (\left(5^{1 / 3}\right)\(\))² = (\((\sqrt[3]{5})\))² ≈ 2.92
a. \(8^{2 / 3}\)
b. \(6^{5 / 2}\)
c. \(12^{3 / 4}\)
d. \(10^{3 / 2}\)
e. \(16^{3 / 2}\)
f. \(20^{6 / 5}\)
Answer:
a. \(8^{2 / 3}\) = (\((\sqrt[3]{8})\))²
The cube root of 8 is 2.
= 2² = 4
b. \(6^{5 / 2}\) = (\((\sqrt[5]{6})\))²
The fifth root of 6 is 1.43
= (1.43)² = 2.04
c. \(12^{3 / 4}\)
The cube root of 12 is 2.28
= (\((\sqrt[3]{12})\))⁴ = (2.28)⁴ = 27.02
d. \(10^{3 / 2}\)
The cube root of 10 is 2.15
= (\((\sqrt[3]{10})\))^² =(2.15)² = 4.62
e. \(16^{3 / 2}\)
The cube root of 16 is 2.51
= (\((\sqrt[3]{16})\))^² =  (2.51)² = 6.3
f. \(20^{6 / 5}\)
The sixth root of 20 is 1.82
= (\((\sqrt[6]{20})\))⁵ =  (1.82)⁵ = 19.96

Communicate Your Answer

Question 4.
How can you use a rational exponent to represent a power involving a radical?
Answer:
In order to rewrite a radical using a fractional exponent, the power to which the radicand becomes the numerator and the root becomes the denominator. Any power can be written in the fractional exponent form.

Question 5.
Evaluate each expression without using a calculator. Explain your reasoning.
a. \(4^{3 / 2}\)
b. \(32^{4 / 5}\)
c. \(625^{3 / 4}\)
d. \(49^{3 / 2}\)
e. \(125^{4 / 3}\)
f. \(100^{6 / 3}\)
Answer:
a. \(4^{3 / 2}\)
= 4^1/2
= 2
= 2³ = 8
So, \(4^{3 / 2}\) = 8
b. \(32^{4 / 5}\)
= 32^1/5×4
= 2^5(1/5×4)
= (2)⁴ = 16
So, \(32^{4 / 5}\) = 16
c. \(625^{3 / 4}\)
= 625^1/4×3
= (625)^1/4 = 5
= (5)³ = 125
d. \(49^{3 / 2}\)
= 49^1/2 = 7
= (7)³= 343
So, \(49^{3 / 2}\) = 343
e. \(125^{4 / 3}\)
= 125^1/3
= (5)⁴ = 625
So, \(125^{4 / 3}\) = 625
f. \(100^{6 / 3}\)
= 100^1/3
= 4.64
= (4.64)⁶ = 9979.
So, \(100^{6 / 3}\) = 9979

5.1 Lesson

Monitoring Progress

Question 1.
n = 4, a = 16
Answer:

Question 2.
n = 2, a = −49
Answer:

Question 3.
n = 3, a = −125
Answer:

 

 

 

 

Question 4.
n = 5, a = 243
Answer:

Evaluate the expression without using a calculator.

Question 5.
\(4^{5 / 2}\)
Answer:

Question 6.
\(9^{-1 / 2}\)
Answer:

Question 7.
\(81^{3 / 4}\)
Answer:

Question 8.
\(1^{7 / 8}\)
Answer:

Evaluate the expression using a calculator. Round your answer to two decimal places when appropriate.

Question 9.
\(6^{2 / 5}\)
Answer:

Question 10.
\(64^{-2 / 3}\)
Answer:

Question 11.
\((\sqrt[4]{16})^{5}\)
Answer:

Question 12.
\((\sqrt[3]{-30})^{2}\)
Answer:

Find the real solution(s) of the equation. Round your answer to two decimal places when appropriate.

Question 13.
8x³ = 64
Answer:

Question 14.
\([\frac{1}{2}/latex]x⁵ = 512
Answer:

Question 15.
(x + 5)⁴ = 16
Answer:

Question 16.
(x − 2)³ = −14
Answer:

Question 17.
WHAT IF? In Example 5, what is the annual depreciation rate when the salvage value is $6000?
Answer:

nth Roots and Rational Exponents 5.1 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY Rewrite the expression [latex]a^{-s / t}\) in radical form. Then state the index of the radical.
Answer:

Question 2.
COMPLETE THE SENTENCE For an integer n greater than 1, if bⁿ = a, then b is a(n) ___________ of a.
Answer:

Question 3.
WRITING Explain how to use the sign of a to determine the number of real fourth roots of a and the number of real fifth roots of a.
Answer:

Question 4.
WHICH ONE DOESN’T BELONG? Which expression does not belong with the other three? Explain your reasoning.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 5–10, find the indicated real nth root(s) of a.

Question 5.
n = 3, a = 8
Answer:

Question 6.
n = 5, a = −1
Answer:

Question 7.
n = 2, a = 0
Answer:

Question 8.
n = 4, a = 256
Answer:

Question 9.
n = 5, a = −32
Answer:

Question 10.
n = 6, a = −729
Answer:

In Exercises 11–18, evaluate the expression without using a calculator.

Question 11.
\(64^{1 / 6}\)
Answer:

Question 12.
\(8^{1 / 3}\)
Answer:

Question 13.
\(25^{3 / 2}\)
Answer:

Question 14.
\(81^{3 / 4}\)
Answer:

Question 15.
\((-243)^{1 / 5}\)
Answer:

Question 16.
\((-64)^{4 / 3}\)
Answer:

Question 17.
\(8^{-2 / 3}\)
Answer:

Question 18.
\(16^{-7 / 4}\)
Answer:

ERROR ANALYSIS In Exercises 19 and 20, describe and correct the error in evaluating the expression.

Question 19.

Answer:

Question 20.

Answer:

USING STRUCTURE In Exercises 21–24, match the equivalent expressions. Explain your reasoning.

Question 21.
\((\sqrt[3]{5})^{4}\) A. \(5^{-1 / 4}\)
Answer:

Question 22.
\((\sqrt[4]{5})^{3}\) B. \(5^{4 / 3}\)
Answer:

Question 23.
\(\frac{1}{\sqrt[4]{5}}\) C. \(-5^{1 / 4}\)
Answer:

Question 24.
\(-\sqrt[4]{5}\) D. \(5^{3 / 4}\)
Answer:

In Exercises 25–32, evaluate the expression using a calculator. Round your answer to two decimal places when appropriate.

Question 25.
\(\sqrt[5]{32,768}\)
Answer:

Question 26.
\(\sqrt[7]{1695}\)
Answer:

Question 27.
\(25^{-1 / 3}\)
Answer:

Question 28.
\(85^{1 / 6}\)
Answer:

Question 29.
\(20,736^{4 / 5}\)
Answer:

Question 30.
\(86^{-5 / 6}\)
Answer:

Question 31.
\((\sqrt[4]{187})^{3}\)
Answer:

Question 32.
\((\sqrt[5]{-8})^{8}\)
Answer:

MATHEMATICAL CONNECTIONS In Exercises 33 and 34, find the radius of the figure with the given volume.

Question 33.
V = 216 ft³

Answer:

Question 34.
V = 1332 cm³

Answer:

In Exercises 35–44, find the real solution(s) of the equation. Round your answer to two decimal places when appropriate.

Question 35.
x³ = 125
Answer:

Question 36.
5x³ = 1080
Answer:

Question 37.
(x + 10)⁵ = 70
Answer:

Question 38.
(x − 5)⁴ = 256
Answer:

Question 39.
x⁵ = −48
Answer:

Question 40.
7x⁴ = 56
Answer:

Question 41.
x⁶ + 36 = 100
Answer:

Question 42.
x³ + 40 = 25
Answer:

Question 43.
\(\frac{1}{3}\)x⁴ = 27
Answer:

Question 44.
\(\frac{1}{6}\)x³ = −36
Answer:

Question 45.
MODELING WITH MATHEMATICS When the average price of an item increases from p₁ to p₂ over a period of n years, the annual rate of inflation r (in decimal form) is given by r = \(\left(\frac{p_{2}}{p_{1}}\right)^{1 / n}\) − 1. Find the rate of inflation for each item in the table.

Answer:

Question 46.
HOW DO YOU SEE IT? The graph of y = xⁿ is shown in red. What can you conclude about the value of n? Determine the number of real nth roots of a. Explain your reasoning.

Answer:

Question 47.
NUMBER SENSE Between which two consecutive integers does \(\sqrt[4]{125}\) lie? Explain your reasoning.
Answer:

Question 48.
THOUGHT PROVOKING In 1619, Johannes Kepler published his third law, which can be given by d³ = t², where d is the mean distance (in astronomical units) of a planet from the Sun and t is the time (in years) it takes the planet to orbit the Sun. It takes Mars 1.88 years to orbit the Sun. Graph a possible location of Mars. Justify your answer. (The diagram shows the Sun at the origin of the xy-plane and a possible location of Earth.)

Answer:

Question 49.
PROBLEM SOLVING A weir is a dam that is built across a river to regulate the flow of water. The flow rate Q (in cubic feet per second) can be calculated using the formula Q= 3.367ℓ\(h^{3 / 2}\), where ℓ is the length (in feet) of the bottom of the spillway and his the depth (in feet) of the water on the spillway. Determine the flow rate of a weir with a spillway that is 20 feet long and has a water depth of 5 feet.

Answer:

Question 50.
REPEATED REASONING The mass of the particles that a river can transport is proportional to the sixth power of the speed of the river. A certain river normally flows at a speed of 1 meter per second. What must its speed be in order to transport particles that are twice as massive as usual? 10 times as massive? 100 times as massive?
Answer:

Maintaining Mathematical Proficiency

Simplify the expression. Write your answer using only positive exponents. (Skills Review Handbook)

Question 51.
5 • 5⁴

Answer:

Question 52.
\(\frac{4^{2}}{4^{7}}\)
Answer:

Question 53.
\(\left(z^{2}\right)^{-3}\)
Answer:

Question 54.
\(\left(\frac{3 x}{2}\right)^{4}\)
Answer:

Write the number in standard form. (Skills Review Handbook)

Question 55.
5 × 10³

Answer:

Question 56.
4 × 10⁻²
Answer:

Question 57.
8.2 × 10⁻¹
Answer:

Question 58.
6.93 × 10⁶
Answer:

Lesson 5.2 Properties of Rational Exponents and Radicals

Essential Question
How can you use properties of exponents to simplify products and quotients of radicals?

EXPLORATION 1
Reviewing Properties of Exponents
Work with a partner. Let a and b be real numbers. Use the properties of exponents to complete each statement. Then match each completed statement with the property it illustrates.

Answer:

Explanation:
a. Negative Exponent: The negative exponent tells how many times we have to divide the base number.
b. Power of Product: The power of a product states that if a term is being to an exponent, you can distribute the power to each factor.
c. Power of a power: Multiply powers together when raising a power by another exponent.
d. Product of Powers: Add powers together when multiplying like bases.
e. Power of a Quotient: Distribute power to all values in a quotient.
f. Quotient of powers: Subtract powers when dividing the bases.
g. Zero Exponent: Any base raised to the power of 0, it gives the result as 1.

EXPLORATION 2
Simplifying Expressions with Rational Exponents
Work with a partner. Show that you can apply the properties of integer exponents to rational exponents by simplifying each expression. Use a calculator to check your answers.
a. \(5^{2 / 3}\) • \(5^{4 / 3}\)
b. \(3^{1 / 5}\) • \(3^{4 / 5}\)
c. \(\left(4^{2 / 3}\right)^{3}\)
d. \(\frac{\sqrt{98}}{\sqrt{2}}\)
e. \(\frac{\sqrt[4]{4}}{\sqrt[4]{1024}}\)
f. \(\frac{\sqrt[3]{625}}{\sqrt[3]{5}}\)

Answer:
a. \(5^{2 / 3}\) • \(5^{4 / 3}\) = 25
Explanation:
Given,
\(5^{2 / 3}\) • \(5^{4 / 3}\)
When bases are equal powers should be added.
5^{2/3+4/3 = 6/3 = 2}
5² = 25
b. \(3^{1 / 5}\) • \(3^{4 / 5}\) = 3
Explanation:
Given,
\(3^{1 / 5}\) • \(3^{4 / 5}\)
=3^{1/5+4/5 = 5/5 = 1}
=3
c. \(\left(4^{2 / 3}\right)^{3}\) = 16
Explanation:
Given,
\(\left(4^{2 / 3}\right)^{3}\)
4^{2/3×3 = 2}
=4²
= 16
d. \(\frac{\sqrt{98}}{\sqrt{2}}\) = 7
Explanation:
Given,
\(\frac{\sqrt{98}}{\sqrt{2}}\)
√(98/2) = √49 = 7
e. \(\frac{\sqrt[4]{4}}{\sqrt[4]{1024}}\) = 5
Explanation:
Given,
\(\frac{\sqrt[4]{4}}{\sqrt[4]{1024}}\)
\(\frac{\sqrt[4]}\)(4/1024) = \(\frac{\sqrt[4]}\)1/231
f. \(\frac{\sqrt[3]{625}}{\sqrt[3]{5}}\)
\(\frac{\sqrt[3]{625}}{\sqrt[3]{5}}\)
= 5

EXPLORATION 3
Simplifying Products and Quotients of Radicals
Work with a partner. Use the properties of exponents to write each expression as a single radical. Then evaluate each expression. Use a calculator to check your answers.

a. \(\sqrt{3}\) • \(\sqrt{12}\)
b. \(\sqrt[3]{5}\) • \(\sqrt[3]{25}\)
c. \(\sqrt[4]{27}\) • \(\sqrt[4]{3}\)
d. \(\frac{\sqrt{98}}{\sqrt{2}}\)
e. \(\frac{\sqrt[4]{4}}{\sqrt[4]{1024}}\)
f. \(\frac{\sqrt[3]{625}}{\sqrt[3]{5}}\)
Answer:
a.
Given,
\(\sqrt{3}\) • \(\sqrt{12}\)
\(\sqrt{a}\) • \(\sqrt{b}\) = \(\sqrt{ab}\)
= \(\sqrt{36}\)
= 6
b.
Given,
\(\sqrt[3]{5}\) • \(\sqrt[3]{25}\)
\(\sqrt{a}\) • \(\sqrt{b}\) = \(\sqrt{ab}\)
= \(\sqrt[3]{125}\)
= 5
c.
Given,
\(\sqrt[4]{27}\) • \(\sqrt[4]{3}\)
\(\sqrt{a}\) • \(\sqrt{b}\) = \(\sqrt{ab}\)
\(\sqrt[4]{81}\)
= 3
d.
Given,
\(\frac{\sqrt{98}}{\sqrt{2}}\)
√(98/2) = √49 = 7
e. \(\frac{\sqrt[4]{4}}{\sqrt[4]{1024}}\)
\(\frac{\sqrt[4]}\)(4/1024) = \(\frac{\sqrt[4]}\)1/231
f.
Given,
\(\frac{\sqrt[3]{625}}{\sqrt[3]{5}}\)
\(\frac{\sqrt[3]{625}}{\sqrt[3]{5}}\) = 5

Communicate Your Answer

Question 4.
How can you use properties of exponents to simplify products and quotients of radicals?
Answer: In order to rewrite radical expression applying first factor of expression which is under root making perfect power.
Property A: x^m . xⁿ = x^m+n
Example:
5² . 5² = 5⁴
Property B: a^m/bⁿ = a^m-n
5²/5¹ = 5^2-1= 5
Property C:
(a^m)ⁿ = a^mn
(5²)¹ = 5²

Question 5.
Simplify each expression.
a. \(\sqrt{27}\) • \(\sqrt{6}\)
Answer:
Given,
\(\sqrt{27}\) • \(\sqrt{6}\)
\(\sqrt{162}\)
b. \(\frac{\sqrt[3]{240}}{\sqrt[3]{15}}\)
Answer:
Given,
\(\frac{\sqrt[3]{240}}{\sqrt[3]{15}}\)
\(\frac{\sqrt[3]{240/15}\) = 16
c. (\(5^{1 / 2}\) \(16^{1 / 4}\))²
Answer:
Given,
(\(5^{1 / 2}\) \(16^{1 / 4}\))²
(\(5^{1 / 2}\) \(16^{1 / 4}\))²
5 × \(16^{1 / 2}\)
5 × 4 = 20

5.2 Lesson

Monitoring Progress

Simplify the expression.

Question 1.
\(2^{3 / 4}\) • \(2^{1 / 2}\)
Answer:
Given,
\(2^{3 / 4}\) • \(2^{1 / 2}\)
x^m . xⁿ = x^m+n
2^{3/4+1/2 = 5/4}
2^5/4
So, \(2^{3 / 4}\) • \(2^{1 / 2}\) = 2^5/4

Question 2.
\(\frac{3}{3^{1 / 4}}\)
Answer:

Question 3.
\(\left(\frac{20^{1 / 2}}{5^{1 / 2}}\right)^{3}\)
Answer:

Question 4.
(\(5^{1 / 3}\) • \(7^{1 / 4}\))³

Answer:

 

Simplify the expression

Question 5.
\(\sqrt[4]{27}\) • \(\sqrt[4]{3}\)
Answer:

Question 6.
\(\frac{\sqrt[3]{250}}{\sqrt[3]{2}}\)
Answer:

Question 7.
\(\sqrt[3]{104}\)
Answer:

Question 8.
\(\sqrt[5]{\frac{3}{4}}\)
Answer:

Question 9.
\(\frac{3}{6-\sqrt{2}}\)
Answer:

Question 10.
\(7 \sqrt[5]{12}\) – \(\sqrt[5]{12}\)
Answer:

Question 11.
4(\(9^{2 / 3}\)) + (\(9^{2 / 3}\))
Answer:

Question 12.
\(\sqrt[3]{5}\) + \(\sqrt[3]{40}\)
Answer:

Simplify the expression. Assume all variables are positive.

Question 13.
\(\sqrt[3]{27 q^{9}}\)
Answer:

Question 14.
\(\sqrt[5]{\frac{x^{10}}{y^{5}}}\)
Answer:

Question 15.
\(\frac{6 x y^{3 / 4}}{3 x^{1 / 2} y^{1 / 2}}\)
Answer:

Question 16.
\(\sqrt{9} w^{5}\) – \(w \sqrt{w^{3}}\)
Answer:

Properties of Rational Exponents and Radicals 5.2 Exercises

Vocabulary and Core Concept Check

Question 1.
WRITING How do you know when a radical expression is in simplest form?
Answer:

Question 2.
WHICH ONE DOESN’T BELONG? Which radical expression does not belong with the other three? Explain your reasoning.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–12, use the properties of rational exponents to simplify the expression.

Question 3.
\(\left(9^{2}\right)^{1 / 3}\)
Answer:

Question 4.
\(\left(12^{2}\right)^{1 / 4}\)
Answer:

Question 5.
\(\frac{6}{6^{1 / 4}}\)
Answer:

Question 6.
\(\frac{7}{7^{1 / 3}}\)
Answer:

Question 7.
\(\left(\frac{8^{4}}{10^{4}}\right)^{-1 / 4}\)
Answer:

Question 8.
\(\left(\frac{9^{3}}{6^{3}}\right)^{-1 / 3}\)
Answer:

Question 9.
(\(3^{-2 / 3}\) • \(3^{1 / 3}\))^-1
Answer:

Question 10.
(\(5^{1 / 2}\) • \(5^{-3 / 2}\))^-1/4
Answer:

Question 11.
\(\frac{2^{2 / 3} \cdot 16^{2 / 3}}{4^{2 / 3}}\)
Answer:

Question 12.
\(\frac{49^{3 / 8} \cdot 49^{7 / 8}}{7^{5 / 4}}\)
Answer:

In Exercises 13–20, use the properties of radicals to simplify the expression.

Question 13.
\(\sqrt{2}\) • \(\sqrt{72}\)
Answer:

Question 14.
\(\sqrt[3]{16}\) • \(\sqrt[3]{32}\)
Answer:

Question 15.
\(\sqrt[4]{6}\) • \(\sqrt[4]{8}\)
Answer:

Question 16.
\(\sqrt[4]{8}\) • \(\sqrt[4]{8}\)
Answer:

Question 17.
\(\frac{\sqrt[5]{486}}{\sqrt[5]{2}}\)
Answer:

Question 18.
\(\frac{\sqrt{2}}{\sqrt{32}}\)
Answer:

Question 19.
\(\frac{\sqrt[3]{6} \cdot \sqrt[3]{72}}{\sqrt[3]{2}}\)
Answer:

Question 20.
\(\frac{\sqrt[3]{3} \cdot \sqrt[3]{18}}{\sqrt[6]{2} \cdot \sqrt[6]{2}}\)
Answer:

In Exercises 21–28, write the expression in simplest form.

Question 21.
\(\sqrt[4]{567}\)
Answer:

Question 22.
\(\sqrt[5]{288}\)
Answer:

Question 23.
\(\frac{\sqrt[3]{5}}{\sqrt[3]{4}}\)
Answer:

Question 24.
\(\frac{\sqrt[4]{4}}{\sqrt[4]{27}}\)
Answer:

Question 25.
\(\sqrt{\frac{3}{8}}\)
Answer:

Question 26.
\(\sqrt[3]{\frac{7}{4}}\)
Answer:

Question 27.
\(\sqrt[3]{\frac{64}{49}}\)
Answer:

Question 28.
\(\sqrt[4]{\frac{1296}{25}}\)
Answer:

In Exercises 29–36, write the expression in simplest form.

Question 29.
\(\frac{1}{1+\sqrt{3}}\)
Answer:

Question 30.
\(\frac{1}{2+\sqrt{5}}\)
Answer: √5 + 2

Explanation:
Given,
\(\frac{1}{2+\sqrt{5}}\) × \(\frac{2-\sqrt{5}{2-\sqrt{5}}\)
= \(\frac{2-\sqrt{5}{4-5}\)
= 2-√5/-1
= √5 + 2
Thus, \(\frac{1}{2+\sqrt{5}}\) = √5 + 2

Question 31.
\(\frac{5}{3-\sqrt{2}}\)
Answer:

Question 32.
\(\frac{11}{9-\sqrt{6}}\)
Answer:

Question 33.
\(\frac{9}{\sqrt{3}+\sqrt{7}}\)
Answer:

Question 34.
\(\frac{2}{\sqrt{8}+\sqrt{7}}\)
Answer:

Question 35.
\(\frac{\sqrt{6}}{\sqrt{3}-\sqrt{5}}\)
Answer:

Question 36.
\(\frac{\sqrt{7}}{\sqrt{10}-\sqrt{2}}\)
Answer:

In Exercises 37–46, simplify the expression.

Question 37.
\(9 \sqrt[3]{11}\) + \(3 \sqrt[3]{11}\)
Answer:

Question 38.
\(8 \sqrt[6]{5}\) – \(12 \sqrt[6]{5}\)
Answer:

Question 39.
\(3\left(11^{1 / 4}\right)\) + \(9\left(11^{1 / 4}\right)\)
Answer:

Question 40.
\(13\left(8^{3 / 4}\right)\) – \(4\left(8^{3 / 4}\right)\)
Answer:

Question 41.
\(5 \sqrt{12}\) – \(19 \sqrt{3}\)
Answer:

Question 42.
\(27 \sqrt{6}\) + \(7 \sqrt{150}\)
Answer: 34\(\sqrt{6}\)

Explanation:
Given,
\(27 \sqrt{6}\) + \(7 \sqrt{6.25}\)
= \(27 \sqrt{6}\) + \(35 \sqrt{6}\)
Take \( \sqrt{6}\) as common factor.
= \( \sqrt{6}\)(27+7)
= 34\(\sqrt{6}\)
So, \(27 \sqrt{6}\) + \(7 \sqrt{150}\) = 34\(\sqrt{6}\)

Question 43.
\(\sqrt[5]{224}\) + \(3 \sqrt[5]{7}\)
Answer:

Question 44.
\(7 \sqrt[3]{2}\) – \(\sqrt[3]{128}\)
Answer:
Given,
\(7 \sqrt[3]{2}\) – \(\sqrt[3]{128}\)
\(\sqrt[3]{128}\) = 2\(\sqrt[3]{16}\)
\(7 \sqrt[3]{2}\) – 2\(\sqrt[3]{16}\)
So, \(7 \sqrt[3]{2}\) – \(\sqrt[3]{128}\) = \(7 \sqrt[3]{2}\) – 2\(\sqrt[3]{16}\)

Question 45.
\(5\left(24^{1 / 3}\right)\) – 4(\(\left(3^{1 / 3}\right)\))
Answer:

Question 46.
\(5^{1 / 4}\) + 6(\([405^{1 / 4}/latex])
Answer:

Question 47.
ERROR ANALYSIS Describe and correct the error in simplifying the expression.

Answer:

Question 48.
MULTIPLE REPRESENTATIONS Which radical expressions are like radicals?
A. [latex]\left(5^{2 / 9}\right)^{3 / 2}\)
B. \(\frac{5^{3}}{(\sqrt[3]{5})^{8}}\)
C. \(\sqrt[3]{625}\)
D. \(\sqrt[3]{5} 145\) – \(\sqrt[3]{875}\)
E. \(\sqrt[3]{5}\) + \(3 \sqrt[3]{5}\)
F. \(7 \sqrt[4]{80}\) – \(2 \sqrt[4]{405}\)
Answer:

In Exercises 49–54, simplify the expression.

Question 49.
\(\sqrt[4]{81 y^{8}}\)
Answer:

Question 50.
\(\sqrt[3]{64 r^{3} t^{6}}\)
Answer:
Given,
\(\sqrt[3]{64 r^{3} t^{6}}\)
= 4r\(\sqrt{t^{6}}\)

Question 51.
\(\sqrt[5]{\frac{m^{10}}{n^{5}}}\)
Answer:

Question 52.
\(\sqrt[4]{\frac{k^{16}}{16 z^{4}}}\)
Answer:

Question 53.
\(\sqrt[6]{\frac{g^{6} h}{h^{7}}}\)
Answer:

Question 54.
\(\sqr

Question 55.
ERROR ANALYSIS Describe and correct the error in simplifying the expression.

Answer:

Question 56.
OPEN-ENDED Write two variable expressions involving radicals, one that needs absolute value in simplifying and one that does not need absolute value. Justify your answers.
Answer:

In Exercises 57–64, write the expression in simplest form. Assume all variables are positive.

Question 57.
[latex]\sqrt{81 a^{7} b^{12} c^{9}}\)
Answer:

Question 58.
\(\sqrt[3]{125 r^{4} s^{9} t^{7}}\)
Answer:

Question 59.
\(\sqrt[5]{\frac{160 m^{6}}{n^{7}}}\)
Answer:

Question 60.
\(\sqrt[4]{\frac{405 x^{3} y^{3}}{5 x^{-1} y}}\)
Answer:

Question 61.
\(\frac{\sqrt[3]{w} \cdot \sqrt{w^{5}}}{\sqrt{25 w^{16}}}\)
Answer:

Question 62.
\(\frac{\sqrt[4]{v^{6}}}{\sqrt[7]{v^{5}}}\)
Answer:​

Question 63.
\(\frac{18 w^{1 / 3} v^{5 / 4}}{27 w^{4 / 3} v^{1 / 2}}\)
Answer:

Question 64.
\(\frac{7 x^{-3 / 4} y^{5 / 2} z^{-2 / 3}}{56 x^{-1 / 2} y^{1 / 4}}\)
Answer:

In Exercises 65–70, perform the indicated operation. Assume all variables are positive.

Question 65.
\(12 \sqrt[3]{y}\) + \(9 \sqrt[3]{y}\)
Answer:

Question 66.
\(11 \sqrt{2 z}\) – \(5 \sqrt{2 z}\)
Answer:
Given,
\(11 \sqrt{2 z}\) – \(5 \sqrt{2 z}\)
Taking \( \sqrt{2 z}\) as common
(11 – 5)\( \sqrt{2 z}\)
6\( \sqrt{2 z}\)
So, \(11 \sqrt{2 z}\) – \(5 \sqrt{2 z}\) = 6\( \sqrt{2 z}\)

Question 67.
\(3 x^{7 / 2}\) – 5\(x^{7 / 2}\)
Answer:

Question 68.
\(7 \sqrt[3]{m^{7}}\) + \(3 m^{7 / 3}\)
Answer:
Given,
\(7 \sqrt[3]{m^{7}}\) + \(3 m^{7 / 3}\)
\(7 m^{7 / 3}\) + \(3 m^{7 / 3}\)
Taking \(m^{7 / 3}\) as common.
(7 + 3)\(m^{7 / 3}\)
10\(m^{7 / 3}\)
So, \(7 \sqrt[3]{m^{7}}\) + \(3 m^{7 / 3}\) = 10\(m^{7 / 3}\)

Question 69.
\(\sqrt[4]{16 w^{10}}\) + \(2 w \sqrt[4]{w^{6}}\)
Answer:

Question 70.
(p^1/2 • p^1/4) – \(\sqrt[4]{16 p^{3}}\)
Answer:

MATHEMATICAL CONNECTIONS In Exercises 71 and 72, find simplified expressions for the perimeter and area of the given figure.

Question 71.

Answer:

Question 72

Answer:

Question 73.
MODELING WITH MATHEMATICS The optimum diameter d (in millimeters) of the pinhole in a pinhole camera can be modeled by d = 1.9[(5.5 × 10⁻⁴)ℓ]^1/2, where ℓ is the length (in millimeters) of the camera box. Find the optimum pinhole diameter for a camera box with a length of 10 centimeters.

Answer:

Question 74.
MODELING WITH MATHEMATICS The surface area S(in square centimeters) of a mammal can be modeled by S = km^2/3, where m is the mass (in grams) of the mammal and k is a constant. The table shows the values of k for different mammals.

a. Find the surface area of a bat whose mass is 32 grams.
b. Find the surface area of a rabbit whose mass is 3.4 kilograms (3.4 × 10³ grams).
c. Find the surface area of a human whose mass is 59 kilograms.
Answer:

Question 75.
MAKING AN ARGUMENT Your friend claims it is not possible to simplify the expression 7\(\sqrt{11}\) − 9 \(\sqrt{44}\) because it does not contain like radicals. Is your friend correct? Explain your reasoning.
Answer:

Question 76.
PROBLEM SOLVING The apparent magnitude of a star is a number that indicates how faint the star is in relation to other stars. The expression f(x) = \(\frac{2.512^{m_{1}}}{2.512^{m_{2}}}\) tells how many times fainter a star with apparent magnitude m₁ is than a star with apparent magnitude m₂.

a. How many times fainter is Altair than Vega?
b. How many times fainter is Deneb than Altair?
c. How many times fainter is Deneb than Vega?

Answer:

Question 77.
CRITICAL THINKING Find a radical expression for the perimeter of the triangle inscribed in the square shown. Simplify the expression.

Answer:

Question 78.
HOW DO YOU SEE IT? Without finding points, match the functions f(x) = \(\sqrt{64 x^{2}}\) and g(x) = \(\sqrt[3]{64 x^{6}}\) with their graphs. Explain your reasoning.

Answer:

Question 79.
REWRITING A FORMULA You have filled two round balloons with water. One balloon contains twice as much water as the other balloon.
a. Solve the formula for the volume of a sphere, V = \(\frac{4}{3}\)πr³, for r.
b. Substitute the expression for r from part (a) into the formula for the surface area of a sphere, S = 4πr². Simplify to show that S = (4π)^1/3(3V)^2/3.
c. Compare the surface areas of the two water balloons using the formula in part (b).
Answer:

Question 80.
THOUGHT PROVOKING Determine whether the expressions (x²)^1/6 and (x^1/6)² are equivalent for all values of x.
Answer:

Question 81.
DRAWING CONCLUSIONS Substitute different combinations of odd and even positive integers for m and n in the expression \(\sqrt[n]{x^{m}}\). When you cannot assume x is positive, explain when absolute value is needed in simplifying the expression.
Answer:

Maintaining Mathematical Proficiency

Identify the focus, directrix, and axis of symmetry of the parabola. Then graph the equation. (Section 2.3)

Question 82.
y = 2x²
Answer:

Question 83.
y² = −x
Answer:

Question 84.
y² = 4x
Answer:

Write a rule for g. Describe the graph of g as a transformation of the graph of f. (Section 4.7)

Question 85.
f(x) = x⁴ − 3x² − 2x, g(x) = −f(x)
Answer:

Question 86.
f(x) = x³ − x, g(x) = f(x) − 3
Answer:

Question 87.
f(x) = x³ − 4, g(x) = f(x − 2)
Answer:

Question 88.
f(x) = x⁴ + 2x³ − 4x², g(x) = f(2x)
Answer:

Lesson 5.3 Graphing Radical Functions

Essential Question

How can you identify the domain and range of a radical function?

EXPLORATION 1
Identifying Graphs of Radical Functions
Work with a partner. Match each function with its graph. Explain your reasoning. Then identify the domain and range of each function.
a. f(x) = \(\sqrt{x}\)
b. f(x) = \(\sqrt[3]{x}\)
c. f(x) = \(\sqrt[4]{x}\)
d. f(x) = \(\sqrt[5]{x}\)

Answer:
a.
f(x) = \(\sqrt{x}\) suits with graph D.
So, D is the correct answer.
b. f(x) = \(\sqrt[3]{x}\) suits with graph C.
So, Graph C is the correct answer.
c. f(x) = \(\sqrt[4]{x}\) suits with graph A.
So, Graph A is the correct answer.
d. f(x) = \(\sqrt[5]{x}\) suits with graph B.
So, Graph B is the correct answer.

EXPLORATION 2
Identifying Graphs of Transformations
Work with a partner. Match each transformation of f(x) = \(\sqrt{x}\) with its graph. Explain your reasoning. Then identify the domain and range of each function.
a. g(x) = \(\sqrt{x+2}\)
b. g(x) = \(\sqrt{x-2}\)
c. g(x) = \(\sqrt{x}+2-2\)
d. g(x) = −\(\sqrt{x+2}\)

Communicate Your Answer

Question 3.
How can you identify the domain and range of a radical function?
Answer:

• The domain and range of a radical function f(x) having an even order are all positive numbers and zero as the radicand takes only positive values.
• The domain and range of a radical function f(x) having an odd order are all real numbers as radicand can take the value of any real number. f(x) = √x.

Question 4.
Use the results of Exploration 1 to describe how the domain and range of a radical function are related to the index of the radical.
Answer:

• The functions \(\sqrt{x}\) and \(\sqrt[4]{x}\) have the domain and the range as [0, ∞)
• The functions \(\sqrt[3]{x}\) and \(\sqrt[5]{x}\) have the domain and the range as (-∞, ∞)
• If the index of the given radical is odd, its domain and range will not have any restriction. Whereas if it is even, they will have restrictions.

5.3 Lesson

Monitoring Progress

Question 1.
Graph g(x) = \(\sqrt{x+1}\). Identify the domain and range of the function.
Answer:

Question 2.
Describe the transformation of f(x) = \(\sqrt[3]{x}\) represented by g(x) = −\(\sqrt[3]{x}\) − 2. Then graph each function.
Answer:

Question 3.
WHAT IF? In Example 3, the function N(d ) = 2.4 • E(d) approximates the number of seconds it takes a dropped object to fall d feet on the Moon. Write a rule for N. How long does it take a dropped object to fall 25 feet on the Moon?
Answer:

Question 4.
In Example 4, is the transformed function the same when you perform the translation followed by the horizontal shrink? Explain your reasoning.
Answer:

Question 5.
Use a graphing calculator to graph −4y² = x + 1. Identify the vertex and the direction that the parabola opens.
Answer:

Question 6.
Use a graphing calculator to graph x² + y² = 25. Identify the radius and the intercepts.
Answer:

Graphing Radical Functions 5.3 Exercises

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE Square root functions and cube root functions are examples of __________ functions.
Answer:
Square root functions and cube root functions are examples of radical functions.

Question 2.
COMPLETE THE SENTENCE When graphing y = a\(\sqrt[3]{x-h}\) + k, translate the graph of y = a\(\sqrt[3]{x}\)h units __________ and k units __________.
Answer:
Square root functions and cube root functions are examples of radical functions

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, match the function with its graph.

Question 3.
f(x) = \(\sqrt{x}+3\)
Answer:

Question 4.
h(x) = \(\sqrt{x}\) + 3
Answer:

Question 5.
f(x) = \(\sqrt{x-3}\)
Answer:

Question 6.
g(x) = \(\sqrt{x}\) − 3
Answer:

Question 7.
h(x) = \(\sqrt{x+3}\) − 3
Answer:

Question 8.
f(x) = \(\sqrt{x-3}\) + 3
Answer:

In Exercises 9–18, graph the function. Identify the domain and range of the function.

Question 9.
h(x) = \(\sqrt{x}\) + 4
Answer:

Question 10.
g(x) = \(\sqrt{x}\) − 5
Answer:

Question 11.
g(x) = − \(\sqrt[3]{2 x}\)
Answer:

Question 12.
f(x) = \(\sqrt[3]{-5 x}\)
Answer:

Question 13.
g(x) = \(\frac{1}{5} \sqrt{x}-3\)
Answer:

Question 14.
f(x) = \(\frac{1}{2} \sqrt[3]{x}+6\)
Answer:

Question 15.
f(x) = \((6 x)^{1 / 2}\) + 3
Answer:

Question 16.
g(x) = −3(x + 1)^1/3

Answer:

Question 17.
h(x) = −\(\sqrt[4]{x}\)
Answer:

Question 18.
h(x) = \(\sqrt[5]{2 x}\)
Answer:

In Exercises 19–26, describe the transformation of f represented by g. Then graph each function.

Question 19.
f(x) = \(\sqrt{x}\), g(x) = \(\sqrt{x+1}\) + 8
Answer:

Question 20.
f(x) = \(\sqrt{x}\), g(x) = 2\(\sqrt{x-1}\)
Answer:

Question 21.
f(x) = \(\sqrt[3]{x}\), g(x) = −\(\sqrt[3]{x}\) − 1
Answer:

Question 22.
f(x) = \(\sqrt[3]{x}\), g(x) = \(\sqrt[3]{x+4}\) − 5
Answer:

Question 23.
f(x) = \(x^{1 / 2}\), g(x) = \(\frac{1}{4}(-x)^{1 / 2}\)
Answer:

Question 24.
f(x) = \(x^{1 / 3}\), g(x) = \(\frac{1}{3} x^{1 / 3}\) + 6
Answer:

Question 25.
f(x) = \(\sqrt[4]{x}\), g(x) = \(2 \sqrt[4]{x+5}\) − 4
Answer:

Question 26.
f(x) = \(\sqrt[5]{x}\), g(x) = \(\sqrt[5]{-32 x}\) + 3
Answer:

Question 27.
ERROR ANALYSIS Describe and correct the error in graphing f(x) = \(\sqrt{x-2}\) − 2.

Answer:

Question 28.
ERROR ANALYSIS Describe and correct the error in describing the transformation of the parent square root function represented by g(x) = \(\sqrt{\frac{1}{2} x}\) + 3.

Answer:
The graph of g is a horizontal stretch by a factor of 2 and translation 3 units up of the parent square root function.

USING TOOLS In Exercises 29–34, use a graphing calculator to graph the function. Then identify the domain and range of the function.

Question 29.
g(x) = \(\sqrt{x^{2}+x}\)
Answer:

Question 30.
h(x) = \(\sqrt{x^{2}-2 x}\)
Answer:

Question 31.
f(x) = \(\sqrt[3]{x^{2}+x}\)
Answer:

Question 32.
f(x) = \(\sqrt[3]{3 x^{2}-x}\)
Answer:

Question 33.
f(x) = \(\sqrt{2 x^{2}+x+1}\)
Answer:

Question 34.
h(x) = \(\sqrt[3]{\frac{1}{2} x^{2}-3 x+4}\)
Answer:

ABSTRACT REASONING In Exercises 35–38, complete the statement with sometimes, always, or never.

Question 35.
The domain of the function y = a\(\sqrt{x}\) is ______ x ≥ 0.
Answer: The domain of the function y = a\(\sqrt{x}\) is always x ≥ 0.

Question 36.
The range of the function y = a\(\sqrt{x}\) is ______ y ≥ 0.
Answer: The range of the function y = a\(\sqrt{x}\) is sometimes y ≥ 0.

Question 37.
The domain and range of the function y = \(\sqrt[3]{x-h}\) + k are ________ all real numbers.
Answer: The domain and range of the function y = \(\sqrt[3]{x-h}\) + k are always all real numbers.

Question 38.
The domain of the function y = a\(\sqrt{-x}\) + k is ________ x ≥ 0.
Answer: The domain of the function y = a\(\sqrt{-x}\) + k is never x ≥ 0.

Question 39.
PROBLEM SOLVING The distance (in miles) a pilot can see to the horizon can be approximated by E(n) = 1.22\(\sqrt{n}\), where n is the plane’s altitude (in feet above sea level) on Earth. The function M(n) = 0.75E(n) approximates the distance a pilot can see to the horizon n feet above the surface of Mars. Write a rule for M. What is the distance a pilot can see to the horizon from an altitude of 10,000 feet above Mars?

Answer:

Question 40.
MODELING WITH MATHEMATICS The speed (in knots) of sound waves in air can be modeled by v(K) = 643.855\(\sqrt{\frac{K}{273.15}}\) where K is the air temperature (in kelvin). The speed (in meters per second) of sound waves in air can be modeled by s(K) = \(\frac{v(K)}{1.944}\). Write a rule for s. What is the speed (in meters per second) of sound waves when the air temperature is 305 kelvin?
Answer:

In Exercises 41–44, write a rule for g described by the transformations of the graph of f.

Question 41.
Let g be a vertical stretch by a factor of 2, followed by a translation 2 units up of the graph of f(x) = \(\sqrt{x}\) + 3.
Answer:

Question 42.
Let g be a reflection in the y-axis, followed by a translation 1 unit right of the graph of f(x) = \(2 \sqrt[3]{x-1}\).
Answer:

Question 43.
Let g be a horizontal shrink by a factor of \(\frac{2}{3}\), followed by a translation 4 units left of the graph of f(x) = \(\sqrt{6 x}\).
Answer:

Question 44.
Let g be a translation 1 unit down and 5 units right, followed by a reflection in the x-axis of the graph of f(x) = −\(-\frac{1}{2} \sqrt[4]{x}+\frac{3}{2}\)
Answer:

In Exercises 45 and 46, write a rule for g.

Question 45.

Answer:

Question 46.

Answer:

In Exercises 47–50, write a rule for g that represents the indicated transformation of the graph of f.

Question 47.
f(x) = 2\(\sqrt{x}\), g(x) = f(x + 3)
Answer:

Question 48.
f(x) = \(\frac{1}{3} \sqrt{x-1}\), g(x) = −f(x) + 9
Answer:

Question 49.
f(x) = −\(\sqrt{x^{2}-2}\), g(x) = −2f(x + 5)
Answer:

Question 50.
f(x) = \(\sqrt[3]{x^{2}+10 x}\), g(x) =\(\frac{1}{4}\)f(−x) + 6
Answer:

In Exercises 51–56, use a graphing calculator to graph the equation of the parabola. Identify the vertex and the direction that the parabola opens.

Question 51.
\(\frac{1}{4}\)y² = x
Answer:

Question 52.
3y² = x
Answer:

Question 53.
−8y² + 2 = x
Answer:

Question 54.
2y² = x − 4
Answer:

Question 55.
x + 8 = \(\frac{1}{5}\)y²
Answer:

Question 56.
\(\frac{1}{2}\)x = y² − 4
Answer:

In Exercises 57–62, use a graphing calculator to graph the equation of the circle. Identify the radius and the intercepts.

Question 57.
x² + y² = 9
Answer:

Question 58.
x² + y² = 4
Answer:

Question 59.
1 − y² = x²
Answer:

Question 60.
64 − x² = y²
Answer:

Question 61.
−y² = x² − 36
Answer:

Question 62.
x² = 100 − y²
Answer:

Question 63.
MODELING WITH MATHEMATICS The period of a pendulum is the time the pendulum takes to complete one back-and-forth swing. The period T (in seconds) can be modeled by the function T = 1.11\(\sqrt{\ell}\), where ℓ is the length (in feet) of the pendulum. Graph the function. Estimate the length of a pendulum with a period of 2 seconds. Explain your reasoning.

Answer:

Question 64.
HOW DO YOU SEE IT? Does the graph represent a square root function or a cube root function? Explain. What are the domain and range of the function?

Answer:

Question 65.
PROBLEM SOLVING For a drag race car with a total weight of 3500 pounds, the speed s(in miles per hour) at the end of a race can be modeled by s = 14.8\(\sqrt[3]{p}\), where p is the power (in horsepower). Graph the function.
a. Determine the power of a 3500-pound car that reaches a speed of 200 miles per hour.
b. What is the average rate of change in speed as the power changes from 1000 horsepower to 1500 horsepower?
Answer:

Question 66.
THOUGHT PROVOKING The graph of a radical function f passes through the points (3, 1) and (4, 0). Write two different functions that could represent f(x + 2) + 1. Explain.
Answer:

Question 67.
MULTIPLE REPRESENTATIONS The terminal velocity v_t(in feet per second) of a skydiver who weighs 140 pounds is given by v_t = 33.7\(\sqrt{\frac{140}{\Lambda}}\)
where A is the cross-sectional surface area (in square feet) of the skydiver. The table shows the terminal velocities (in feet per second) for various surface areas (in square feet) of a skydiver who weighs 165 pounds.

a. Which skydiver has a greater terminal velocity for each value of A given in the table?
b. Describe how the different values of A given in the table relate to the possible positions of the falling skydiver.
Answer:

Question 68.
MATHEMATICAL CONNECTIONS The surface area S of a right circular cone with a slant height of 1 unit is given by S = πr + πr², where r is the radius of the cone.

a. Use completing the square to show that r = \(\frac{1}{\sqrt{\pi}} \sqrt{S+\frac{\pi}{4}}-\frac{1}{2}\).
b. Graph the equation in part (a) using a graphing calculator. Then find the radius of a right circular cone with a slant height of 1 unit and a surface area of \(\frac{3 \pi}{4}\) square units.
Answer:

Maintaining Mathematical Proficiency

Solve the equation. Check your solutions. (Skills Review Handbook)

Question 69.
| 3x + 2 | = 5
Answer:

Question 70.
∣4x + 9 ∣ = −7
Answer:

Question 71.
| 2x − 6 ∣ = ∣ x ∣
Answer:

Question 72.
| x + 8 | = | 2x + 2 |
Answer:
Given,
| x + 8 | = | 2x + 2 |
x + 8 = 2x + 2
x – 2x = 2 – 8
-x = -6
x = 6
x + 8 = -2x – 2
x + 2x = -2 – 8
3x = -10
x = -10/3

Solve the inequality. (Section 3.6)

Question 73.
x² + 7x + 12 < 0
Answer:

Question 74.
x² − 10x + 25 ≥ 4
Answer:

Question 75.
2x² + 6 > 13x
Answer:

Question 76.
\(\frac{1}{8}\)x² + x ≤ −2
Answer:

Rational Exponents and Radical Functions Study Skills : Analyzing Your Errors

5.1–5.3 What Did You Learn?

Core Vocabulary

Core Concepts
Section 5.1
Real nth Roots of a, p. 238
Rational Exponents, p. 239

Section 5.2
Properties of Rational Exponents, p. 244
Properties of Radicals, p. 245

Section 5.3
Parent Functions for Square Root and Cube Root Functions, p. 252
Transformations of Radical Functions, p. 253

Mathematical Practices

Question 1.
How can you use definitions to explain your reasoning in Exercises 21–24 on page 241?

Question 2.
How did you use structure to solve Exercise 76 on page 250?

Question 3.
How can you check that your answer is reasonable in Exercise 39 on page 257?

Question 4.
How can you make sense of the terms of the surface area formula given in Exercise 68 on page 258?

Study Skills
Analyzing Your Errors
Application Errors
What Happens: You can do numerical problems, but you struggle with problems that have context.

How to Avoid This Error: Do not just mimic the steps of solving an application problem. Explain out loud what the question is asking and why you are doing each step. After solving the problem, ask yourself, “Does my solution make sense?”

Rational Exponents and Radical Functions 5.1 – 5.3 Quiz

5.1–5.3 Quiz

Find the indicated real nth root(s) of a. (Section 5.1)

Question 1.
n = 4, a = 81
Answer: 3 and -3

Explanation:
Given,
\(\sqrt[n]{a}\)
where, n = 4, a = 81
\(\sqrt[4]{81}\)
The nth root is even so its root will be negative or positive.
The root of this is a number when multiplied by itself 4 times will give 81.
81 is odd and is a multiple of 3 and 9 so it is likely 3.
3 × 3 × 3 × 3 = 81
-3 × -3 × -3 × -3 = 81
The real nth root are 3 and -3

Question 2.
n = 5, a = −1024
Answer: -5

Explanation:
Given,
\(\sqrt[5]{-1024}\)
The nth root when n is, even so, the root will be negative or positive.
The root of this is a number when multiplied by itself 5 times will give 1024.
1024 is even and is multiple.
-5 × -5 × -5 × -5 × -5 = -1024
The real nth root of a is -5.

Question 3.
Evaluate (a) 16^3/4 and (b) 125^2/3 without using a calculator. Explain your reasoning. (Section 5.1)
Answer:
(a) 16^3/4
\(\sqrt[4]{16^3}\)
= \(\sqrt[4]{4096}\)
= 8
(b) 125^2/3
\(\sqrt[3]{125^2}\)
\(\sqrt[3]{15625}\)
= 25

Find the real solution(s) of the equation. Round your answer to two decimal places.(Section 5.1)

Question 4.
2x⁶ = 1458
Answer: x = 3

Explanation:
Given,
2x⁶ = 1458
x⁶ = 1458/2
x⁶ = 729
x⁶ = 3⁶
x = 3
The real solution of the equation is x = 3.

Question 5.
(x + 6)³ = 28
Answer: x(x² + 18x + 108) = -188

Explanation:
Given,
(x + 6)³ = 28
This is in the form of (a + b)³ = a³ + b³ + 3a²b + 3ab²
(x)³ + (6)³ + 3(x)²(6) + 3(x)(6)² = 28
x³ + 216 + 18x² + 108x = 28
x³ + 18x² + 108x = 28 – 216
x³ + 18x² + 108x = – 188
x(x² + 18x + 108) = -188
So, the equation is x(x² + 18x + 108) = -188

Simplify the expression.(Section 5.2)

Question 6.
\(\left(\frac{48^{1 / 4}}{6^{1 / 4}}\right)^{6}\)

Question 7.
\(\sqrt[4]{3}\) • \(\sqrt[4]{432}\)

Question 8.
\(\frac{1}{3+\sqrt{2}}\)

Question 9.
\(\sqrt[3]{16}\) – \(5 \sqrt[3]{2}\)

Question 10.
Simplify \(\sqrt[8]{x^{9} y^{8} z^{16}}\). (Section 5.2)

Write the expression in simplest form. Assume all variables are positive.(Section 5.2)

Question 11.
\(\sqrt[3]{216 p^{9}}\)

Question 12.
\(\frac{\sqrt[5]{32}}{\sqrt[5]{m^{3}}}\)

Question 13.
\(\sqrt[4]{n^{4} q}\) + \(7 n \sqrt[4]{q}\)

Question 14.
Graph f(x) = 2\(2 \sqrt[3]{x}\) + 1. Identify the domain and range of the function. (Section 5.3)

Describe the transformation of the parent function represented by the graph of g. Then write a rule for g.(Section 5.3)

Question 15.

Question 16.

Question 17.

Question 18.
Use a graphing calculator to graph x = 3y² − 6. Identify the vertex and direction the parabola opens. (Section 5.3)

Question 19.
A jeweler is setting a stone cut in the shape of a regular octahedron. A regular octahedron is a solid with eight equilateral triangles as faces, as shown. The formula for the volume of the stone is V= 0.47s³, where s is the side length (in millimeters) of an edge of the stone. The volume of the stone is 161 cubic millimeters. Find the length of an edge of the stone. (Section 5.1)

Question 20.
An investigator can determine how fast a car was traveling just prior to an accident using the model s = 4\(\sqrt{d}\), where s is the speed (in miles per hour) of the car and d is the length (in feet) of the skid marks. Graph the model. The length of the skid marks of a car is 90 feet. Was the car traveling at the posted speed limit prior to the accident? Explain your reasoning. (Section 5.3)

Lesson 5.4 Solving Radical Equations and Inequalities

Essential Question
How can you solve a radical equation?

EXPLORATION 1
Solving Radical Equations
Work with a partner. Match each radical equation with the graph of its related radical function. Explain your reasoning. Then use the graph to solve the equation, if possible. Check your solutions.
a. \(\sqrt{x-1}\) – 1 = 0
b. \(\sqrt{2 x+2}\) – \(\sqrt{x+4}\) = 0
c. \(\sqrt{9-x^{2}}\) = 0
d. \(\sqrt{x+2}\) – x = 0
e. \(\sqrt{-x+2}\) – x = 0
f. \(\sqrt{3 x^{2}+1}\) = 0

EXPLORATION 2
Solving Radical Equations
Work with a partner. Look back at the radical equations in Exploration 1. Suppose that you did not know how to solve the equations using a graphical approach.
a. Show how you could use a numerical approach to solve one of the equations. For instance, you might use a spreadsheet to create a table of values.

b. Show how you could use an analytical approach to solve one of the equations. For instance, look at the similarities between the equations in Exploration 1. What first step may be necessary so you could square each side to eliminate the radical(s)? How would you proceed to find the solution?

Communicate Your Answer

Question 3.
How can you solve a radical equation?

Question 4.
Would you prefer to use a graphical, numerical, or analytical approach to solve the given equation? Explain your reasoning. Then solve the equation.
\(\sqrt{x+3}\) – \(\sqrt{x-2}\) = 1

Monitoring Progress

Solve the equation. Check your solution.

Question 1.
\(\sqrt[3]{x}\) – 9 = -6
Answer:

Question 2.
\(\sqrt{x+25}\) = 2
Answer:

Question 3.
2\(\sqrt[3]{x-3}\) = 4
Answer:

Question 4.
WHAT IF? Estimate the air pressure at the center of the hurricane when the mean sustained wind velocity is 48.3 meters per second.
Answer:

Solve the equation. Check your solution(s).

Question 5.
\(\sqrt{10 x+9}\) = x + 3
Answer:

Question 6.
\(\sqrt{2 x+5}\) = \(\sqrt{x+7}\)
Answer:

Question 7.
\(\sqrt{x+6}\) – 2 = \(\sqrt{x-2}\)
Answer:

Solve the equation. Check your solution(s).

Question 8.
(3x)^1/3 = −3
Answer:

Question 9.
(x + 6)^1/2 = x
Answer:

Question 10.
(x + 2)^3/4 = 8
Answer:

Question 11.
Solve
(a) 2\(\sqrt{x}\) − 3 ≥ 3 and
(b) 4\(\sqrt[3]{x+1}\) < 8
Answer:

Solving Radical Equations and Inequalities 5.4 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY Is the equation 3x − \(\sqrt{2}\) = \(\sqrt{6}\) a radical equation? Explain your reasoning.
Answer:

Question 2.
WRITING Explain the steps you should use to solve \(\sqrt{x}\) + 10 < 15
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–12, solve the equation. Check your solution.

Question 3.
\(\sqrt{5 x+1}\) = 6
Answer:

Question 4.
\(\sqrt{3 x+10}\) = 8
Answer:

Question 5.
\(\sqrt[3]{x-16}\) = 2
Answer:

Question 6.
\(\sqrt[3]{x}\) − 10 = −7
Answer:

Question 7.
−2\(\sqrt{24 x}\) + 13 = −1
Answer:

Question 8.
8\(\sqrt[3]{10 x}\) − 15 = 17
Answer:

Question 9.
\(\frac{1}{5} \sqrt[3]{3 x}\) = 8
Answer:

Question 10.
\(\sqrt{2 x}-\frac{2}{3}\) = 0
Answer:

Question 11.
\(2 \sqrt[5]{x}\) + 7 = 15
Answer:

Question 12.
\(\sqrt[4]{4 x}\) − 13 = −15
Answer:

Question 13.
MODELING WITH MATHEMATICS Biologists have discovered that the shoulder height h (in centimeters) of a male Asian elephant can be modeled by h = 62.5\(\sqrt[3]{t}\) + 75.8, where t is the age (in years) of the elephant. Determine the age of an elephant with a shoulder height of 250 centimeters.

Answer:

Question 14.
MODELING WITH MATHEMATICS In an amusement park ride, a rider suspended by cables swings back and forth from a tower. The maximum speed v (in meters per second) of the rider can be approximated by v = \(\sqrt{2 g h}\), where h is the height (in meters) at the top of each swing and g is the acceleration due to gravity (g ≈ 9.8 m/sec²). Determine the height at the top of the swing of a rider whose maximum speed is 15 meters per second.
Answer:

In Exercises 15–26, solve the equation. Check your solution(s).

Question 15.
x− 6 = \(\sqrt{3 x}\)
Answer:

Question 16.
x − 10 = \(\sqrt{9 x}\)
Answer:

Question 17.
\(\sqrt{44-2 x}\) = x − 10
Answer:

Question 18.
\(\sqrt{2 x+30}\) = x + 3
Answer:

Question 19.
\(\sqrt[3]{8 x^{3}-1}\) = 2x − 1
Answer:

Question 20.
\(\sqrt[4]{3-8 x^{2}}\) = 2x
Answer:

Question 21.
\(\sqrt{4 x+1}\) = \(\sqrt{x+10}\)
Answer:

Question 22.
\(\sqrt{3 x-3}\) – \(\sqrt{x+12}\) = 0
Answer:

Question 23.
\(\sqrt[3]{2 x-5}\) – \(\sqrt[3]{8 x+1}\) = 0
Answer:

Question 24.
\(\sqrt[3]{x+5}\) = 2\(\sqrt[3]{2 x+6}\)
Answer:

Question 25.
\(\sqrt{3 x-8}\) + 1 = \(\sqrt{x+5}\)
Answer:

Question 26.
\(\sqrt{x+2}\) = 2 – \(\sqrt{x}\)
Answer:

In Exercises 27–34, solve the equation. Check your solution(s).

Question 27.
2x^2/3 = 8
Answer:

Question 28.
4x^3/2 = 32
Answer:
Given,
4x^3/2 = 32
x = 32/4
x^3/2 = 8
\(\sqrt{x³}\) = 8
Squaring on both sides
x³ = 64
x³ = 4³
x = 4

Question 29.
x^1/4 + 3 = 0
Answer:

Question 30.
2x^3/4 − 14 = 40
Answer:

Question 31.
(x + 6)^1/2 = x
Answer:

Question 32.
(5 − x)^1/2 − 2x = 0
Answer:

Question 33.
2(x + 11)^1/2 = x + 3
Answer:

Question 34.
(5x² − 4)^1/4 = x
Answer:

ERROR ANALYSIS In Exercises 35 and 36, describe and correct the error in solving the equation.

Question 35.

Answer:

Question 36.

Answer:

In Exercises 37–44, solve the inequality.

Question 37.
\(2 \sqrt[3]{x}\) − 5 ≥ 3
Answer:

Question 38.
\(\sqrt[3]{x-4}\) ≤ 5
Answer:
Given,
\(\sqrt[3]{x-4}\) ≤ 5
Apply Cubing on both sides
(x – 4) ≤ 5³
x-4 ≤ 125
x ≤ 125 + 4
x ≤ 129

Question 39.
\(4 \sqrt{x-2}\) > 20
Answer:

Question 40.
7\(\sqrt{x}\) + 1 < 9
Answer:

Question 41.
2\(\sqrt{x}\) + 3 ≤ 8
Answer:

Question 42.
\(\sqrt[3]{x+7}\) ≥ 3
Answer:

Question 43.
\(-2 \sqrt[3]{x+4}\) < 12
Answer:

Question 44.
−0.25\(\sqrt{x}\) − 6 ≤ −3
Answer:

Question 45.
MODELING WITH MATHEMATICS The length ℓ (in inches) of a standard nail can be modeled by ℓ = 54d^3/2, where d is the diameter (in inches) of the nail. What is the diameter of a standard nail that is 3 inches long?
Answer:

Question 46.
DRAWING CONCLUSIONS “Hang time” is the time you are suspended in the air during a jump. Your hang time t (in seconds) is given by the function t = 0.5\(\sqrt{h}\), where h is the height (in feet) of the jump. Suppose a kangaroo and a snowboarder jump with the hang times shown.

a. Find the heights that the snowboarder and the kangaroo jump.
b. Double the hang times of the snowboarder and the kangaroo and calculate the corresponding heights of each jump.
c. When the hang time doubles, does the height of the jump double? Explain.
Answer:

USING TOOLS In Exercises 47–52, solve the nonlinear system. Justify your answer with a graph.

Question 47.
y² = x − 3
y = x − 3
Answer:

Question 48.
y² = 4x + 17
y = x + 5
Answer:

Question 49.
x² + y² = 4
y = x – 2
Answer:

Question 50.
x² + y² = 25
y = \(-\frac{3}{4}\)x + \(\frac{25}{4}\)
Answer:

Question 51.
x² + y² = 1
y = \(\frac{1}{2}\)x² – 1
Answer:

Question 52.
x² + y² = 4
y² = x + 2
Answer:

Question 53.
PROBLEM SOLVING The speed s (in miles per hour) of a car can be given by s = \(\sqrt{30 f d}\), where f is the coefficient of friction and d is the stopping distance (in feet). The table shows the coefficient of friction for different surfaces.

a. Compare the stopping distances of a car traveling 45 miles per hour on the surfaces given in the table.
b. You are driving 35 miles per hour on an icy road when a deer jumps in front of your car. How far away must you begin to brake to avoid hitting the deer? Justify your answer.
Answer:

Question 54.
MODELING WITH MATHEMATICS The Beaufort wind scale was devised to measure wind speed. The Beaufort numbers B, which range from 0 to 12, can be modeled by B = 1.69\(\sqrt{s+4.25}\) − 3.55, where s is the wind speed (in miles per hour).

a. What is the wind speed for B = 0? B = 3?
b. Write an i

Question 55.
USING TOOLS Solve the equation x − 4 = \(\sqrt{2 x}\). Then solve the equation x − 4 = − \(\sqrt{2 x}\).
a. How does changing \(\sqrt{2 x}\) to −\(\sqrt{2 x}\) change the solution(s) of the equation?
b. Justify your answer in part (a) using graphs.
Answer:

Question 56.
MAKING AN ARGUMENT Your friend says it is impossible for a radical equation to have two extraneous solutions. Is your friend correct? Explain your reasoning.
Answer:

Question 57.
USING STRUCTURE Explain how you know the radical equation \(\sqrt{x+4}\) = −5 has no real solution without solving it.
Answer:

Question 58.
HOW DO YOU SEE IT? Use the graph to find the solution of the equation 2\(\sqrt{x-4}\) = −\(\sqrt{x-1}\) + 4. Explain your reasoning.

Answer:

Question 59.
WRITING A company determines that the price p of a product can be modeled by p = 70 − \(\sqrt{0.02 x+1}\), where x is the number of units of the product demanded per day. Describe the effect that raising the price has on the number of units demanded.
Answer:

Question 60.
THOUGHT PROVOKING City officials rope off a circular area to prepare for a concert in the park. They estimate that each person occupies 6 square feet. Describe how you can use a radical inequality to determine the possible radius of the region when P people are expected to attend the concert.

Answer:

Question 61.
MATHEMATICAL CONNECTIONS The Moeraki Boulders along the coast of New Zealand are stone spheres with radii of approximately 3 feet. A formula for the radius of a sphere is
r = \(\frac{1}{2} \sqrt{\frac{S}{\pi}}\)
where S is the surface area of the sphere. Find the surface area of a Moeraki Boulder.
Answer:

Question 62.
PROBLEM SOLVING You are trying to determine the height of a truncated pyramid, which cannot be measured directly. The height h and slant heightℓof the truncated pyramid are related by the formula below.

In the given formula, b1 and b2 are the side lengths of the upper and lower bases of the pyramid, respectively. When ℓ = 5, b₁ = 2, and b₂ = 4, what is the height of the pyramid?
Answer:

Question 63.
REWRITING A FORMULA A burning candle has a radius of r inches and was initially h0inches tall. After t minutes, the height of the candle has been reduced to h inches. These quantities are related by the formula
r = \(\sqrt{\frac{k t}{\pi\left(h_{0}-h\right)}}\)
where k is a constant. Suppose the radius of a candle is 0.875 inch, its initial height is 6.5 inches, and k = 0.04.
a. Rewrite the formula, solving for h in terms of t.
b. Use your formula in part (a) to determine the height of the candle after burning 45 minutes.
Answer:

Maintaining Mathematical Proficiency

Perform the indicated operation. (Section 4.2 and Section 4.3)

Question 64.
(x³ − 2x² + 3x + 1) + (x⁴ − 7x)
Answer:
x⁴ − 7x + x³ − 2x² + 3x + 1
= x⁴ + x³ − 2x² − 7x  + 3x + 1
= x⁴ + x³ − 2x² − 4x + 1
So, (x³ − 2x² + 3x + 1) + (x⁴ − 7x) = x⁴ + x³ − 2x² − 4x + 1

Question 65.
(2x⁵ + x⁴ − 4x²) − (x⁵ − 3)
Answer:

Question 66.
(x³ + 2x² + 1)(x² + 5)
Answer:

Question 67.
(x⁴ + 2x³ + 11x² + 14x − 16) ÷ (x + 2)
Answer:

Let f(x) = x³ – 4x² + 6. Write a rule for g. Describe the graph of g as a transformation of the graph of f.(Section 4.7)

Question 68.
g(x) = f(−x) + 4
Answer:

Question 69.
g(x) = \(\frac{1}{2}\)f(x) − 3
Answer:

Question 70.
g(x) = −f(x − 1) + 6
Answer:

Lesson 5.5 Performing Function Operations

Essential Question
How can you use the graphs of two functions to sketch the graph of an arithmetic combination of the two functions?
Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to form other functions. For example, the functions f(x) = 2x − 3 and g(x) = x² − 1 can be combined to form the sum, difference, product, or quotient of f and g.
f(x) + g(x) = (2x − 3) + (x² − 1) = x² + 2x − 4 sum
f(x) − g(x) = (2x − 3) − (x² − 1) = −x² + 2x − 2 difference
f(x) • g(x) = (2x − 3)(x² − 1) = 2x³ − 3x² − 2x + 3 product

EXPLORATION 1
Graphing the Sum of Two Functions
Work with a partner. Use the graphs of f and g to sketch the graph of f + g. Explain your steps.
Sample Choose a point on the graph of g. Use a compass or a ruler to measure its distance above or below the x-axis. If above, add the distance to the y-coordinate of the point with the same x-coordinate on the graph of f. If below, subtract the distance. Plot the new point. Repeat this process for several points. Finally, draw a smooth curve through the new points to obtain the graph of f + g.

Communicate Your Answer

Question 2.
How can you use the graphs of two functions to sketch the graph of an arithmetic combination of the two functions?

Question 3.
Check your answers in Exploration 1 by writing equations for f and g, adding the functions, and graphing the sum.

5.5 Lesson

Monitoring Progress

Question 1.
Let f(x) = −2x^2/3 and g(x) = 7x^2/3. Find (f + g)(x) and (f − g)(x) and state the domain of each. Then evaluate (f + g)(8) and (f − g)(8).
Answer:

Question 2.
Let f(x) = 3x and g(x) = x^1/5. Find (fg)(x) and (\(\frac{f}{g}\))(x) and state the domain of each. Then evaluate (fg)(32) and (\(\frac{f}{g}\))(32).
Answer:

Question 3.
Let f(x) = 8x and g(x) = 2x^5/6. Use a graphing calculator to evaluate (f + g)(x), (f − g)(x), (fg)(x), and (\(\frac{f}{g}\)) (x) when x = 5. Round your answers to two decimal places.
Answer:

Question 4.
In Example 5, explain why you can evaluate (f + g)(3), (f − g)(3), and (fg)(3) but not (\(\frac{f}{g}\))(3).
Answer:

Question 5.
Use the answer in Example 6(a) to find the total number of heartbeats over the lifetime of a white rhino when its body mass is 1.7 × 10⁵ kilograms.
Answer:

Performing Function Operations 5.5 Exercises

Vocabulary and Core Concept Check

Question 1.
WRITING Let f and g be any two functions. Describe how you can use f, g, and the four basic operations to create new functions.
Answer:

Question 2.
WRITING What x-values are not included in the domain of the quotient of two functions?
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, find (f + g)(x) and (f – g)(x) and state the domain of each. Then evaluate f + g and f – g for the given value of x.

Question 3.
f(x) = \(-5 \sqrt[4]{x}\), g(x) = 19\(\sqrt[4]{x}\); x = 16
Answer:

Question 4.
f(x) = \(\sqrt[3]{2 x}\), g(x) = −11\(\sqrt[3]{2 x}\) ; x = −4
Answer:

Question 5.
f(x) = 6x − 4x²− 7x³, g(x) = 9x²− 5x; x = −1
Answer:

Question 6.
f(x) = 11x + 2x², g(x) = −7x − 3x² + 4; x = 2
Answer:

In Exercises 7–12, find (fg)(x) and (\(\frac{f}{g}\))(x) and state the domain of each. Then evaluate fg and \(\frac{f}{g}\) for the given value of x.

Question 7.
f(x) = 2x³, g(x) = \(\sqrt[3]{x}\) ; x = −27
Answer:

Question 8.
f(x) = x⁴, g(x) = \(3 \sqrt{x}\) ; x = 4
Answer:

Question 9.
f(x) = 4x, g(x) = 9x^1/2; x = 9
Answer:

Question 10.
f(x) = 11x³, g(x) = 7x^7/3; x = −8
Answer:

Question 11.
f(x) = 7x^3/2, g(x) =−14x^1/3; x = 64
Answer:

Question 12.
f(x) = 4x^5/4, g(x) = 2x^1/2; x = 16
Answer:

USING TOOLS In Exercises 13–16, use a graphing calculator to evaluate (f + g)(x), (f − g)(x), (fg)(x), and (\(\frac{f}{g}\))(x) when x = 5. Round your answers to two decimal places.

Question 13.
f(x) = 4x⁴; g(x) = 24x^1/3

Answer:

Question 14.
f(x) = 7x^5/3; g(x) = 49x^2/3
Answer:

Question 15.
f(x) =−2x^1/3; g(x) = 5x^1/2

Answer:

Question 16.
f(x) = 4x^1/2; g(x) = 6x^3/4
Answer:

ERROR ANALYSIS In Exercises 17 and 18, describe and correct the error in stating the domain.

Question 17.

Answer:

Question 18.

Answer:

Question 19.
MODELING WITH MATHEMATICS From 1990 to 2010, the numbers (in millions) of female F and male M employees from the ages of 16 to 19 in the United States can be modeled by F(t) =−0.007t² + 0.10t + 3.7 and M(t) = 0.0001t³ − 0.009t² + 0.11t + 3.7, where t is the number of years since 1990.
a. Find (F + M)(t).
b. Explain what (F + M)(t) represents.
Answer:

Question 20.
MODELING WITH MATHEMATICS From 2005 to 2009, the numbers of cruise ship departures (in thousands) from around the world W and Florida F can be modeled by the equations
W(t) = −5.8333t³ + 17.43t² + 509.1t + 11496
F(t) = 12.5t³ − 60.29t² + 136.6t + 4881
where t is the number of years since 2005.
a. Find (W − F )(t).
b. Explain what (W − F )(t) represents.
Answer:

Question 21.
MAKING AN ARGUMENT Your friend claims that the addition of functions and the multiplication of functions are commutative. Is your friend correct? Explain your reasoning.
Answer:

Question 22.
HOW DO YOU SEE IT? The graphs of the functions f(x) = 3x² − 2x − 1 and g(x) = 3x + 4 are shown. Which graph represents the function f + g? the function f − g? Explain your reasoning.

Answer:

Question 23.
REASONING The table shows the outputs of the two functions f and g. Use the table to evaluate (f + g)(3), (f − g)(1), (fg)(2), and (\(\frac{f}{g}\))(0).

Answer:

Question 24.
THOUGHT PROVOKING Is it possible to write two functions whose sum contains radicals, but whose product does not? Justify your answers.
Answer:

Question 25.
MATHEMATICAL CONNECTIONS A triangle is inscribed in a square, as shown. Write and simplify a function r in terms of x that represents the area of the shaded region.

Answer:

Question 26.
REWRITING A FORMULA For a mammal that weighs w grams, the volume b (in milliliters) of air breathed in and the volume d (in milliliters) of “dead space” (the portion of the lungs not filled with air) can be modeled by
b(w) = 0.007w and d(w) = 0.002w.
The breathing rate r (in breaths per minute) of a mammal that weighs w grams can be modeled by
r(w) = \(\frac{1.1 w^{0.734}}{b(w)-d(w)}\).
Answer:

Question 27.
PROBLEM SOLVING A mathematician at a lake throws a tennis ball from point A along the water’s edge to point B in the water, as shown. His dog, Elvis, first runs along the beach from point A to point D and then swims to fetch the ball at point B.

a. Elvis runs at a speed of about 6.4 meters per second. Write a function r in terms of x that represents the time he spends running from point A to point D. Elvis swims at a speed of about 0.9 meter per second. Write a function s in terms of x that represents the time he spends swimming from point D to point B.
b. Write a function t in terms of x that represents the total time Elvis spends traveling from point A to point D to point B.
c. Use a graphing calculator to graph t. Find the value of x that minimizes t. Explain the meaning of this value.
Answer:

Maintaining Mathematical Proficiency

Solve the literal equation for n. (Skills Review Handbook)

Question 28.
3xn − 9 = 6y
Answer:
Given,
3xn − 9 = 6y
Take 3 as a common factor
xn – 3 = 2y
xn = 2y + 3
n = (2y + 3)/x

Question 29.
5z = 7n + 8nz
Answer:

Question 30.
3nb = 5n − 6z
Answer:
Given,
3nb = 5n − 6z
3nb – 5n = -6z
Take n as the common factor to make n as the subject.
n(3b – 5) = -6z
n = -6z/(3b – 5)

Question 31.
\(\frac{3+4 n}{n}\) = 7b
Answer:

Determine whether the relation is a function. Explain. (Skills Review Handbook)

Question 32.
(3, 4), (4, 6), (1, 4), (2, −1)
Answer:

Question 33.
(−1, 2), (3, 7), (0, 2), (−1, −1)
Answer:

Question 34.
(1, 6), (7, −3), (4, 0), (3, 0)
Answer:

Question 35.
(3, 8), (2, 5), (9, 5), (2, −3)
Answer:

Lesson 5.6 Inverse of a Function

Essential Question
How can you sketch the graph of the inverse of a function?

EXPLORATION 1
Graphing Functions and Their Inverses
Work with a partner. Each pair of functions are inverses of each other. Use a graphing calculator to graph f and g in the same viewing window. What do you notice about the graphs?

a. f(x) = 4x + 3
g(x) = \(\frac{x-3}{4}\)
b. f(x) = x³ + 1
g(x) = \(\sqrt[3]{x-1}\)
c. f(x) = \(\sqrt{x-3}[latex]
g(x) = x² + 3, x ≥ 0
d. f(x) = [latex]\frac{4 x+4}{x+5}\)
g(x) = \(\frac{4-5 x}{x-4}\)

EXPLORATION 2
Sketching Graphs of Inverse Functions
Work with a partner. Use the graph of f to sketch the graph of g, the inverse function of f, on the same set of coordinate axes. Explain your reasoning.

Communicate Your Answer

Question 3.
How can you sketch the graph of the inverse of a function?

Question 4.
In Exploration 1, what do you notice about the relationship between the equations of f and g? Use your answer to find g, the inverse function of
f(x) = 2x − 3.
Use a graph to check your answer.

5.6 Lesson

Monitoring Progress

Solve y = f(x) for x. Then find the input(s) when the output is 2.

Question 1.
f(x) = x − 2
Answer:
Given,
f(x) = x − 2
The values of x are the input values of the function.
f(x) = 2
If the output value of the function f is 2.
2 = x – 2
2 + 2 = x
x = 4

Question 2.
f(x) = 2x²

Answer:
f(x) = 2x²
The values of x are the input values of the function.
f(x) = 2
If the output value of the function f is 2.
2 = 2x²
x² = 1
x = ±1

Question 3.
f(x) = −x³ + 3
Answer:
Given,
f(x) = −x³ + 3
The values of x are the input values of the function.
f(x) = 2
If the output value of the function f is 2.
2 = -x³ + 3
-x³ = 3 – 2
-x³ = 1
-x = ±1

Find the inverse of the function. Then graph the function and its inverse.

Question 4.
f(x) = 2x

Question 5.
f(x) = −x + 1

Question 6.
f(x) = \(\frac{1}{3}\)x − 2

Find the inverse of the function. Then graph the function and its inverse.

Question 7.
f(x) = −x², x ≤ 0

Question 8.
f(x) = −x³ + 4

Question 9.
f(x) = \(\sqrt{x+2}\)

Determine whether the functions are inverse functions.

Question 10.
f(x) = x + 5, g(x) = x − 5

Question 11.
f(x) = 8x³, g(x) = \(\sqrt[3]{2 x}\)

Question 12.
The distance d (in meters) that a dropped object falls in t seconds on Earth is represented by d = 4.9t². Find the inverse of the function. How long does it take an object to fall 50 meters?

Inverse of a Function 5.6 Exercises

Vocabulary and Core Concept Check

Question 1.
VOCABULARY In your own words, state the definition of inverse functions.
Answer:

Question 2.
WRITING Explain how to determine whether the inverse of a function is also a function.
Answer:

Question 3.
COMPLETE THE SENTENCE Functions f and g are inverses of each other provided that f(g(x)) = ____ and g(f(x)) = ____.
Answer:

Question 4.
DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 5–12, solve y = f(x) for x. Then find the input(s) when the output is -3.

Question 5.
f(x) = 3x + 5
Answer:

Question 6.
f(x) = −7x − 2
Answer:

Question 7.
f(x) = \(\frac{1}{2}\)x − 3
Answer:

Question 8.
f(x) = −\(\frac{2}{3}\)x + 1
Answer:

Question 9.
f(x) = 3x³

Answer:

Question 10.
f(x) = 2x⁴ − 5
Answer:

Question 11.
f(x) = (x − 2)² − 7
Answer:

Question 12.
f(x) = (x − 5)³ − 1
Answer:

In Exercises 13–20, find the inverse of the function. Then graph the function and its inverse.

Question 13.
f(x) = 6x
Answer:

Question 14.
f(x) = −3x
Answer:

Question 15.
f(x) = −2x + 5
Answer:

Question 16.
f(x) = 6x − 3
Answer:

Question 17.
f(x) = −\(\frac{1}{2}\)x + 4
Answer:

Question 18.
f(x) = \(\frac{1}{3}\)x − 1
Answer:

Question 19.
f(x) = \(\frac{2}{3}\)x − \(\frac{1}{3}\)
Answer:

Question 20.
f(x) = −\(\frac{4}{5}\)x + \(\frac{1}{5}\)
Answer:

Question 21.
COMPARING METHODS Find the inverse of the function f(x) = −3x + 4 by switching the roles of x and y and solving for y. Then find the inverse of the function f by using inverse operations in the reverse order. Which method do you prefer? Explain.
Answer:

Question 22.
REASONING Determine whether each pair of functions f and g are inverses. Explain your reasoning.

Answer:

In Exercises 23–28, find the inverse of the function. Then graph the function and its inverse.

Question 23.
f(x) = 4x², x ≤ 0
Answer:

Question 24.
f(x) = 9x², x ≤ 0
Answer:

Question 25.
f(x) = (x − 3)³
Answer:

Question 26.
f(x) = (x + 4)³
Answer:

Question 27.
f(x) = 2x⁴, x ≥ 0
Answer:

Question 28.
f(x) = −x⁶, x ≥ 0
Answer:

ERROR ANALYSIS In Exercises 29 and 30, describe and correct the error in finding the inverse of the function.

Question 29.

Answer:

Question 30.

Answer:

USING TOOLS In Exercises 31–34, use the graph to determine whether the inverse of f is a function. Explain your reasoning.

Question 31.

Answer:

Question 32.

Answer:
As per the horizontal line test, the horizontal line intersects the graph more than once. f foes do not have the inverse function. therefore the inverse of the function is not a function.

Question 33.

Answer:

Question 34.

Answer:
As per the horizontal line test, the horizontal line intersects the graph only once, thus the inverse of the function is a function.

In Exercises 35–46, determine whether the inverse of f is a function. Then find the inverse.

Question 35.
f(x) = x³ − 1
Answer:

Question 36.
f(x) = −x³ + 3
Answer:

Question 37.
f(x) = \(\sqrt{x+4}\)
Answer:

Question 38.
f(x) = \(\sqrt{x-6}\)
Answer:

Question 39.
f(x) = \(2 \sqrt[3]{x-5}\)
Answer:

Question 40.
f(x) = 2x² − 5
Answer:

Question 41.
f(x) = x⁴ + 2
Answer:

Question 42.
f(x) = 2x³ − 5
Answer:

Question 43.
f(x) = \(3 \sqrt[3]{x+1}\)
Answer:

Question 44.
f(x) = \(-\sqrt[3]{\frac{2 x+4}{3}}\)
Answer:

Question 45.
f(x) = \(\frac{1}{2}\)x⁵
Answer:

Question 46.
f(x) = \(-3 \sqrt{\frac{4 x-7}{3}}\)
Answer:

Question 47.
WRITING EQUATIONS What is the inverse of the function whose graph is shown?
A. g(x) = \(\frac{3}{2}\)x − 6
B. g(x) = \(\frac{3}{2}\)x + 6
C. g(x) = \(\frac{2}{3}\)x − 6
D. g(x) = \(\frac{2}{3}\)x + 12

Answer:

Question 48.
WRITING EQUATIONS What is the inverse of f(x) = −\(-\frac{1}{64}\)x³?
A. g(x) = −4x³
B. g(x) = 4\(\sqrt[3]{x}\)
C. g(x) = −4\(\sqrt[3]{x}\)
D. g(x) = \(\sqrt[3]{-4 x}\)
Answer:

In Exercises 49–52, determine whether the functions are inverses.

Question 49.
f(x) = 2x − 9, g(x) = x — 2 + 9
Answer:

Question 50.
f(x) = \(\frac{x-3}{4}\), g(x) = 4x + 3
Answer:

Question 51.
f(x) = \(\sqrt[5]{\frac{x+9}{5}}\), g(x) = 5x⁵ − 9
Answer:

Question 52.
f(x) = 7x^3/2 − 4, g(x) = (\(\frac{x+4}{7}\))^3/2

Answer:

Question 53.
MODELING WITH MATHEMATICS The maximum hull speed v (in knots) of a boat with a displacement hull can be approximated by v = 1.34\(\sqrt{\ell}\), where ℓ is the waterline length (in feet) of the boat. Find the inverse function. What waterline length is needed to achieve a maximum speed of 7.5 knots?

Answer:

Question 54.
MODELING WITH MATHEMATICS Elastic bands can be used for exercising to provide a range of resistance. The resistance R (in pounds) of a band can be modeled by R = \(\frac{3}{8}\)L − 5, where L is the total length (in inches) of the stretched band. Find the inverse function. What length of the stretched band provides 19 pounds of resistance?

Answer:

ANALYZING RELATIONSHIPS In Exercises 55–58, match the graph of the function with the graph of its inverse.

Question 55.

Answer:

Question 56.

Answer:

Question 57.

Answer:

Question 58.

Answer:

Question 59.
REASONING You and a friend are playing a number-guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 2, and then add 3. Your friend’s final answer is 53. What was the original number chosen? Justify your answer.
Answer:

Question 60.
MAKING AN ARGUMENT Your friend claims that every quadratic function whose domain is restricted to nonnegative values has an inverse function. Is your friend correct? Explain your reasoning.
Answer:

Question 61.
PROBLEM SOLVING When calibrating a spring scale, you need to know how far the spring stretches for various weights. Hooke’s Law states that the length a spring stretches is proportional to the weight attached to it. A model for one scale is ℓ = 0.5w + 3, whereℓ is the total length (in inches) of the stretched spring and w is the weight (in pounds) of the object.

a. Find the inverse function. Describe what it represents.
b. You place a melon on the scale, and the spring stretches to a total length of 5.5 inches. Determine the weight of the melon.
c. Verify that the function ℓ = 0.5w + 3 and the inverse model in part (a) are inverse functions.
Answer:

Question 62.
THOUGHT PROVOKING Do functions of the form y = x^m/n, where m and n are positive integers, have inverse functions? Justify your answer with examples.
Answer:

Question 63.
PROBLEM SOLVING At the start of a dog sled race in Anchorage, Alaska, the temperature was 5°C. By the end of the race, the temperature was −10°C. The formula for converting temperatures from degrees Fahrenheit F to degrees Celsius C is C = \(\frac{5}{9}\)(F − 32).
a. Find the inverse function. Describe what it represents.
b. Find the Fahrenheit temperatures at the start and end of the race.

c. Use a graphing calculator to graph the original function and its inverse. Find the temperature that is the same on both temperature scales.
Answer:

Question 64.
PROBLEM SOLVING The surface area A (in square meters) of a person with a mass of 60 kilograms can be approximated by A = 0.2195h^0.3964, where h is the height (in centimeters) of the person.
a. Find the inverse function. Then estimate the height of a 60-kilogram person who has a body surface area of 1.6 square meters.
b. Verify that function A and the inverse model in part (a) are inverse functions.
Answer:

USING STRUCTURE In Exercises 65–68, match the function with the graph of its inverse.

Question 65.
f(x) = \(\sqrt[3]{x-4}\)
Answer:

Question 66.
f(x) = \(\sqrt[3]{x+4}\)
Answer:

Question 67.
f(x) = \(\sqrt{x+1}\) – 3
Answer:

Question 68.
f(x) = \(\sqrt{x-1}\) + 3
Answer:

Question 69.
DRAWING CONCLUSIONS Determine whether the statement is true or false. Explain your reasoning.
a. If f(x) = xⁿ and n is a positive even integer, then the inverse of f is a function.
b. If f(x) = xⁿ and n is a positive odd integer, then the inverse of f is a function.
Answer:

Question 70.
HOW DO YOU SEE IT? The graph of the function f is shown. Name three points that lie on the graph of the inverse of f. Explain your reasoning.

Answer:

Question 71.
ABSTRACT REASONING Show that the inverse of any linear function f(x) = mx + b, where m ≠ 0, is also a linear function. Identify the slope and y-intercept of the graph of the inverse function in terms of m and b.
Answer:

Question 72.
CRITICAL THINKING Consider the function f(x) = −x.
a. Graph f(x) = −x and explain why it is its own inverse. Also, verify that f(x) = −x is its own inverse algebraically.
b. Graph other linear functions that are their own inverses. Write equations of the lines you graphed.
c. Use your results from part (b) to write a general equation describing the family of linear functions that are their own inverses.
Answer:

Maintaining Mathematical Proficiency

Simplify the expression. Write your answer using only positive exponents.(Skills Review Handbook)

Question 73.
(−3)⁻³

Answer:

Question 74.
2³ • 2²
Answer:
Given
2³ • 2²
When bases are equal powers should be added.
2³⁺² = 2⁵

Question 75.
\(\frac{4^{5}}{4^{3}}\)
Answer:

Question 76.
(\(\frac{2}{3}\))⁴
Answer:
Given,
(\(\frac{2}{3}\))⁴
(\(\frac{a}{b}\))ⁿ= aⁿ/bⁿ
= 2⁴/3⁴
= 16/81

Describe the x-values for which the function is increasing, decreasing, positive, and negative.

Question 77.

Answer:

Question 78.

Answer:

Question 79.

Answer:

Rational Exponents and Radical Functions Performance Task: Turning the Tables

5.4–5.6 What Did You Learn?

Core Vocabulary
radical equation, p. 262
extraneous solutions, p. 263
inverse functions, p. 277

Core Concepts
Section 5.4
Solving Radical Equations, p. 262
Solving Radical Inequalities, p. 265

Section 5.5
Operations on Functions, p. 270

Section 5.6
Exploring Inverses of Functions, p. 276
Inverses of Nonlinear Functions, p. 278
Horizontal Line Test, p. 278

Mathematical Practices

Question 1.
How did you find the endpoints of the range in part (b) of Exercise 54 on page 267?

Question 2.
How did you use structure in Exercise 57 on page 268?

Question 3.
How can you evaluate the reasonableness of the results in Exercise 27 on page 274?

Question 4.
How can you use a graphing calculator to check your answers in Exercises 49–52 on page 282?

Performance Task

Turning the Tables

In this chapter, you have used properties of rational exponents and functions to find an answer to the problem. Using those same properties, can you find a problem to the answer? How many problems can you find?

To explore the answers to these questions and more, go to BigIdeasMath.com

Rational Exponents and Radical Functions Chapter Review

Evaluate the expression without using a calculator.

Question 1.
8^7/3

Answer:
Given,
8^7/3
We can write the given radical as
8⁷ . 8^1/3
= 2097152 × 2
= 4194304
So, 8^7/3 = 4194304

Question 2.
9^5/2

Answer:
Given,
9^5/2
We can write the given radical as
9⁵ × 9^1/2
= 59049 × 3
= 177147
So, 9^5/2 = 177147

Question 3.
(−27)^−2/3

Answer:
Given,
(-27)⁻² × (-27)^1/3
= -3 × √27

Find the real solution(s) of the equation. Round your answer to two decimal places when appropriate.

Question 4.
x⁵ + 17 = 35
Answer:
Given,
x⁵ + 17 = 35
x⁵ = 35 – 17
x⁵ = 18
Applying the fifth root on both sides
x = 1.79

Question 5.
7x³ = 189
Answer:
Given,
7x³ = 189
x³ = 189/7
x³ = 27
Cubing on both sides
x = 3

Question 6.
(x + 8)⁴ = 16
Answer:
Given,
(x + 8)⁴ = 16
16 can be written as 2⁴
(x + 8)⁴ = 2⁴
x + 8 = 2
x = 2 – 8
x = -6

Simplify the expression.

Question 7.
(\(\frac{6^{1 / 5}}{6^{2 / 5}}\))³

Answer:
(\(\frac{6^{1 / 5}}{6^{2 / 5}}\))³
(\(\frac{6^{3 / 5}}{6^{6 / 5}}\))

Question 8.
\(\sqrt[4]{32} \cdot \sqrt[4]{8}\)
Answer:
Given,
\(\sqrt[4]{32} \cdot \sqrt[4]{8}\)
= \(\sqrt[4]{32×8}\)
= \(\sqrt[4]{256}\)
= \(\sqrt[4]{4^4}\)
= 4

Question 9.
\(\frac{1}{2-\sqrt[4]{9}}\)
Answer:
\(\frac{1}{2-\sqrt[4]{9}}\)
\(\sqrt[2]{9}\) = 1.73
= 1/(2 – 1.73)
= 1/0.27
= 3.7

Question 10.
\(4 \sqrt[5]{8}\) + \(3 \sqrt[5]{8}\)
Answer:
Given,
\(4 \sqrt[5]{8}\) + \(3 \sqrt[5]{8}\)
= \(\sqrt[5]{8}\)(4 + 3)
= 7\(\sqrt[5]{8}\)
So, \(4 \sqrt[5]{8}\) + \(3 \sqrt[5]{8}\) = 7\(\sqrt[5]{8}\)

Question 11.
\(2 \sqrt{48}\) – \(\sqrt{3}\)
Answer:
Given,
\(2 \sqrt{48}\) – \(\sqrt{3}\)
48 = 16 × 3
\(2 \sqrt{16×3}\) – \(\sqrt{3}\)
2 × 4 \(\sqrt{3}\) – \(\sqrt{3}\)
\(\sqrt{3}\)(8 – 1)
= 7\(\sqrt{3}\)
\(2 \sqrt{48}\) – \(\sqrt{3}\) = 7\(\sqrt{3}\)

Question 12.
(\(5^{2 / 3} \cdot 2^{3 / 2}\))^1/2

Answer:
Given,
(5^2/3)^1/2 × (2^3/2)^1/2
=5^1/3 × 2^3/4
= \(\sqrt[3]{5}\) . \(\sqrt[4]{8}\)
= 2\(\sqrt[3]{5}\) . \(\sqrt[2]{2}\)
So, (\(5^{2 / 3} \cdot 2^{3 / 2}\))^1/2 =2\(\sqrt[3]{5}\) . \(\sqrt[2]{2}\)

Simplify the expression. Assume all variables are positive.

Question 13.
\(\sqrt[3]{125 z^{9}}\)

Question 14.
\(\frac{2^{1 / 4} z^{5 / 4}}{6 z}\)

Question 15.
\(\sqrt{10 z^{5}}-z^{2} \sqrt{40 z}\)

Describe the transformation of f represented by g. Then graph each function.

Question 16.
f(x) = \(\sqrt{x}\), g(x) = −2\(\sqrt{x}\)

Question 17.
f(x) = \(\sqrt[3]{x}\), g(x) = \(\sqrt[3]{-x}\) − 6

Question 18.
Let the graph of g be a reflection in the y-axis, followed by a translation 7 units to the right of the graph of f(x) = \(\sqrt[3]{x}\). Write a rule for g.

Question 19.
Use a graphing calculator to graph 2y² = x − 8. Identify the vertex and the direction that the parabola opens.

Question 20.
Use a graphing calculator to graph x² + y² = 81. Identify the radius and the intercepts.

Solve the equation. Check your solution.

Question 21.
\(4 \sqrt[3]{2 x+1}\) = 20

Question 22.
\(\sqrt{4 x-4}\) = \(\sqrt{5 x-1}\) – 1

Question 23.
(6x)^2/3 = 36

Solve the inequality.

Question 24.
\(5 \sqrt{x}\) + 2 > 17

Question 25.
\(2 \sqrt{x-8}\) < 24

Question 26.
7\(\sqrt[3]{x-3}\) ≥ 21

Question 27.
In a tsunami, the wave speeds (in meters per second) can be modeled by s(d ) = \(\sqrt{9.8 d}\), where d is the depth (in meters) of the water. Estimate the depth of the water when the wave speed is 200 meters per second.

Question 28.
Let f(x) = 2\(\sqrt{3-x}\) and g(x) = 4\(\sqrt[3]{3-x}\). Find (fg)(x) and (\(\frac{f}{g}\))(x) and state the domain of each. Then evaluate(fg)(2) and (\(\frac{f}{g}\))(2).

Question 29.
Let f(x) = 3x² + 1 and g(x) = x + 4. Find (f + g)(x) and (f − g)(x) and state the domain of each. Then evaluate (f + g)(−5) and (f − g)(−5).

Question 30.
f(x) = −\(\frac{1}{2}\)x + 10

Question 31.
f(x) = x² + 8, x ≥ 0

Question 32.
f(x) = −x³ − 9

Question 33.
f(x) = 3\(\sqrt{x}\) + 5

Determine whether the functions are inverse functions.

Question 34.
f(x) = 4(x − 11)², g(x) = \(\frac{1}{4}\)(x + 11)²

Question 35.
f(x) = −2x + 6, g(x) = −\(\frac{1}{2}\)x + 3

Question 36.
On a certain day, the function that gives U.S. dollars in terms of British pounds is d = 1.587p, where d represents U.S. dollars and p represents British pounds. Find the inverse function. Then find the number of British pounds equivalent to 100 U.S. dollars.

Rational Exponents and Radical Functions Chapter Test

Question 1.
Solve the inequality \(\) − 2 ≤ 13 and the equation 5\(\) − 2 = 13. Describe the similarities and differences in solving radical equations and radical inequalities.

Describe the transformation of f represented by g. Then write a rule for g.

Question 2.
f(x) = \(\sqrt{x}\)

Question 3.
f(x) = \(\sqrt[3]{x}\)

Question 4.
f(x) = \(\sqrt[5]{x}\)

Simplify the expression. Explain your reasoning.

Question 5.
64^2/3

Question 6.
(−27)^5/3

Question 7.
\(\sqrt[4]{48 x y^{11} z^{3}}\)

Question 8.
\(\frac{\sqrt[3]{256}}{\sqrt[3]{32}}\)

Question 9.
Write two functions whose graphs are translations of the graph of y = \(\sqrt{x}\). The first function should have a domain of x ≥ 4. The second function should have a range of y ≥ −2.

Question 10.
In bowling, a handicap is a change in score to adjust for differences in the abilities of players. You belong to a bowling league in which your handicap h is determined using the formula h = 0.9(200 − a), where a is your average score. Find the inverse of the model. Then find the average for a bowler whose handicap is 36.

Question 11.
The basal metabolic rate of an animal is a measure of the amount of calories burned at rest for basic functioning. Kleiber’s law states that an animal’s basal metabolic rate R (in kilocalories per day) can be modeled by R = 73.3w^3/4, where w is the mass (in kilograms) of the animal. Find the basal metabolic rates of each animal in the table.

Question 12.
Let f(x) = 6x^3/5 and g(x) = −x^3/5. Find (f + g)(x) and (f − g)(x) and state the domain of each. Then evaluate (f + g)(32) and (f − g)(32).

Question 13.
Let f(x) = \(\frac{1}{2}\)x^3/4 and g(x) = 8x. Find (fg)(x) and (\(\frac{f}{g}\))(x) and state the domain of each. Then evaluate (fg)(16) and (\(\frac{f}{g}\))(16).

Question 14.
A football player jumps to catch a pass. The maximum height h (in feet) of the player above the ground is given by the function h = \(\frac{1}{64}\)s², where s is the initial speed (in feet per second) of the player. Find the inverse of the function. Use the inverse to find the initial speed of the player shown. Verify that the functions are inverse functions.

Rational Exponents and Radical Functions Cumulative Assessment

Question 1.
Identify three pairs of equivalent expressions. Assume all variables are positive. Justify your answer.

Question 2.
The graph represents the function . Choose the correct values to complete the function.

Question 3.
In rowing, the boat speed s (in meters per second) can be modeled by s= 4.62\(\sqrt[9]{n}\), where n is the number of rowers.
a. Find the boat speeds for crews of 2 people, 4 people, and 8 people.
b. Does the boat speed double when the number of rowers doubles? Explain.
c. Find the time (in minutes) it takes each crew in part (a) to complete a 2000-meter race.

Question 4.
A polynomial function fits the data in the table. Use finite differences to find the degree of the function and complete the table. Explain your reasoning.

Question 5.
The area of the triangle is 42 square inches. Find the value of x.

Answer:
Given,
The area of the triangle is 42 square inches.
b = (x + 8)
h = x
We know that,
Area of the triangle = 1/2 × b × h
A = 42
42 = 1/2 × (x + 8) × x
84 = x² + 8x
x² + 8x – 84 = 0

Question 6.
Which equations are represented by parabolas? Which equations are functions? Place check marks in the appropriate spaces. Explain your reasoning.

Question 7.
What is the solution of the inequality 2\(\sqrt{x+3}\) − 1 < 3?
A. x < 1
B. −3 < x < 1
C. −3 ≤ x < 1
D. x ≥ −3

Question 8.
Which function does the graph represent? Explain your reasoning.

Question 9.
Your friend releases a weather balloon 50 feet from you. The balloon rises vertically. When the balloon is at height h, the distance d between you and the balloon is given by d = \(\sqrt{2500+h^{2}}\), where h and d are measured in feet. Find the inverse of the function. What is the height of the balloon when the distance between you and the balloon is 100 feet?

Question 10.
The graphs of two functions f and g are shown. Are f and g inverse functions? Explain your reasoning.

Are you looking for the best guide for the questions avail in Algebra 2 Data Analysis and Statistics? If yes, then no more searching, here you can find Big Ideas Math Algebra 2 Answer Key Ch 11 Data Analysis and Statistics. All the students can get detailed solutions provided for all the BIM Algebra 2 Chapter 11 Data Analysis and Statistics Questions ensuring they understand the concepts simply. So, Download Big Ideas Math Algebra 2 Answers in order to learn the concepts in Chapter 11 Data Analysis and Statistics in depth.

Big Ideas Math Book Algebra 2 Answer Key Chapter 11 Data Analysis and Statistics

Check out the topics covered in Big Ideas Math Textbook Algebra 2 Chapter 11 Data Analysis and Statistics before you start your preparation. All the questions are prepared as per the Big Ideas Math Algebra 2 Textbooks. Here the students can get the solutions for Exercises, Chapter Tests, Review Tests, Cumulative Assessments, Practice Tests, etc. Click on the links available below to start your preparation. Also, you can test your knowledge by solving the questions in Quizzes, Chapter Reviews and Chapter tests and know how much skills you have improved from here.

• Data Analysis and Statistics Maintaining Mathematical Proficiency – Page 593
• Data Analysis and Statistics Mathematical Practices – Page 594
• Lesson 11.1 Using Normal Distributions – Page(597-602)
• Using Normal Distributions 11.1 Exercises – Page(600 – 602)
• Lesson 11.2 Populations, Samples, and Hypotheses – Page(603 – 608)
• Populations, Samples, and Hypotheses 11.2 Exercises – Page(607 – 608)
• Lesson 11.3 Collecting Data – Page(609 – 616)
• Collecting Data 11.3 Exercises – Page(614 – 616)
• Data Analysis and Statistics Study Skills: Reworking Your Notes – Page 617
• Data Analysis and Statistics 11.1–11.3 Quiz – Page 618
• Lesson 11.4 Experimental Design – Page(619 – 624)
• Experimental Design 11.4 Exercises – Page(623 – 624)
• Lesson 11.5 Making Inferences from Sample Surveys – Page(625 – 632)
• Making Inferences from Sample Surveys 11.5 Exercises – Page(630 – 632)
• Lesson 11.6 Making Inferences from Experiments – Page(633 – 638)
• Making Inferences from Experiments 11.6 Exercises – Page(637 – 638)
• Data Analysis and Statistics Performance Task: Curving the Test – Page 639
• Data Analysis and Statistics Chapter Review – Page(640 – 642)
• Data Analysis and Statistics Chapter Test – Page 643
• Data Analysis and Statistics Cumulative Assessment – Page(644 – 646)

Data Analysis and Statistics Maintaining Mathematical Proficiency

Find the mean, median, and mode of the data set. Then determine which measure of center best represents the data. Explain.
Question 1.
36, 82, 94, 83, 86, 82
Answer:
Given data is 36, 82, 94, 83, 86, 82
Mean is the average of the numbers.
Mean = Sum of observations/ no. of observations
Mean = \(\frac { 36 + 82 + 94 + 83 + 86 + 82 }{ 6 } \)
= 77.16
So, Mean = 77.16
The median best represents the data. It is the middle value of the given list of data when arranged in order.
Arrange the data in ascending order.
36, 82, 82, 83, 86, 94
Median = \(\frac { 82 + 83 }{ 2 } \)
= 82.5
So, Median = 82.5
Mode is defined as the most frequent number.
36, 82, 82, 83, 86, 94
82 is repeated 2 times.
So, the mode is 82.
Mode = 82

Question 2.
74, 89, 71, 70, 68, 70
Answer:
Given data is 74, 89, 71, 70, 68, 70
Mean is the average of the numbers.
Mean = Sum of observations/ no. of observations
Mean = \(\frac {74 + 89 + 71 + 70 + 68 + 70  }{ 6 } \)
= 73.6
The median best represents the data. It is the middle value of the given list of data when arranged in order.
Arrange the data in ascending order.
68, 70, 70, 71, 74, 89
Median = \(\frac { 70 + 71 }{ 2 } \)
= 70.5
Mode is defined as the most frequent number.
68, 70, 70, 71, 74, 89
70 is repeated 2 times.
Mode = 70

Question 3.
1, 18, 12, 16, 11, 15, 17, 44, 44
Answer:
Given data is 1, 18, 12, 16, 11, 15, 17, 44, 44
Mean is the average of the numbers.
Mean = Sum of observations/ no. of observations
Mean = \(\frac { 1 + 18 + 12 + 16 + 11 + 15 + 17 + 44 + 44 }{ 9} \)
= 19.7
The median best represents the data. It is the middle value of the given list of data when arranged in order.
Arrange the data in ascending order.
1, 11, 12, 15, 16, 17, 18, 44, 44
Median = 16
Mode is defined as the most frequent number.
44 is repeated 2 times.
Mode = 44

Find and interpret the standard deviation of the data set.
Question 4.
43, 48, 41, 51, 42
Answer:
Given data is 43, 48, 41, 51, 42
Mean is the average of the numbers.
Mean = Sum of observations/ no. of observations
Mean = \(\frac { 43 + 48 + 41 + 51 + 42 }{ 5} \)
= 45
So, mean = 45

x	μ	(x – μ)	(x – μ)²
43	45	-2	4
48	45	3	9
41	45	-4	16
51	45	6	36
42	45	-3	9

Variance = \(\frac { 4 + 9 + 16 + 36 + 9 }{ 5 } \)
= 14.8
Standard deviation = √(14.8) = 3.84

Question 5.
28, 26, 21, 44, 29, 32
Answer:
Given data is 28, 26, 21, 44, 29, 32
Mean is the average of the numbers.
Mean = Sum of observations/ no. of observations
Mean = \(\frac { 28 + 26 + 21 + 44 + 29 + 32 }{ 6} \)
= 30

x	μ	(x – μ)	(x – μ)²
28	30	-2	4
26	30	-4	16
21	30	-9	81
44	30	14	196
29	30	-1	1
32	30	2	4

Variance = \(\frac { 4 + 16 + 81 + 196 + 1 + 4 }{ 6 } \)
= 50.3
Standard deviation = √(50.3) = 7.09

Question 6.
65, 56, 49, 66, 62, 52, 53, 49
Answer:
Given data is 65, 56, 49, 66, 62, 52, 53, 49
Mean is the average of the numbers.
Mean = Sum of observations/ no. of observations
Mean = \(\frac { 65+ 56+ 49+ 66+ 62+ 52+ 53+ 49 }{ 8} \)
= 56.5

x	μ	(x – μ)	(x – μ)²
65	56.5	8.5	72.25
56	56.5	-0.5	0.25
49	56.5	-7.5	56.25
66	56.5	9.5	90.25
62	56.5	5.5	30.25
52	56.5	-4.5	20.25
53	56.5	-3.5	12.25
49	56.5	-7.5	56.25

Variance = \(\frac { 72.25 + 0.25 + 56.25 + 90.25 + 30.25 + 20.25 + 12.25 + 56.25 }{ 8 } \)
= 42.25
Standard deviation = √(42.25) = 6.5

Question 7.
ABSTRACT REASONING
Describe a data set that has a standard deviation of zero. Can a standard deviation be negative? Explain your reasoning.
Answer:
A dataset having a standard deviation of zero would indicate that every data value would be equal to the mean. If all the observations are equal then the standard deviation is 0.
Standard Deviation cannot be negative because when we have at least two numbers in the data set which are not exactly equal to one another, the standard deviation has to be positive.

Data Analysis and Statistics Mathematical Practices

Mathematically proficient students use diagrams and graphs to show relationships between data. They also analyze data to draw conclusions.

Monitoring Progress

Use the Internet or some other reference to determine which age pyramid is that of Canada, Japan, and Mexico. Compare the mean, median, and mode of the three age pyramids.
Question 1.

Answer:
For Male,
Mean is the average of the numbers.
Mean = Sum of observations/ no. of observations
Mean = 30.65347
The median best represents the data. It is the middle value of the given list of data when arranged in order.
Median class = 24.5 – 29.5 = 27.57
Modal class = 4.5 – 9.5
Mode is defined as the most frequent number.

Mode = 8.3274
For Female,
Mean = Sum of observations/ no. of observations
Mean = 32.2377
The median best represents the data. It is the middle value of the given list of data when arranged in order.
Median class = 29.5 – 34.5
Median = 29.7866
Modal class = 14.5-19.5
Mode is defined as the most frequent number.
Mode = 16.7714

Question 2.

Answer:
Given, Pyramid
Mean is the average of the numbers.
Mean = Sum of observations/ no. of observations
a. The mean is less than that of Japan’s but higher than Mexico’s
The median best represents the data. It is the middle value of the given list of data when arranged in order.
Arrange the data in ascending order.
b. Median = 14.8
Mode is defined as the most frequent number
c. Mode = 45 years

Question 3.

Answer:

Lesson 11.1 Using Normal Distributions

Essential Question In a normal distribution, about what percent of the data lies within one, two, and three standard deviations of the mean?
Recall that the standard deviation σ of a numerical data set is given by
σ = \(\sqrt{\frac{\left(x_{1}-\mu\right)^{2}+\left(x_{2}-\mu\right)^{2}+\cdots+\left(x_{n}-\mu\right)^{2}}{n}}\)
where n is the number of values in the data set and μ is the mean of the data set.

EXPLORATION 1

Analyzing a Normal Distribution
Work with a partner. In many naturally occurring data sets, the histogram of the data is bell-shaped. In statistics, such data sets are said to have a normal distribution. For the normal distribution shown below, estimate the percent of the data that lies within one, two, and three standard deviations of the mean. Each square on the grid represents 1%.

EXPLORATION 2

Analyzing a Data Set
Work with a partner. A famous data set was collected in Scotland in the mid-1800s. It contains the chest sizes (in inches) of 5738 men in the Scottish Militia. Do the data fit a normal distribution? Explain.

Communicate Your Answer

Question 3.
In a normal distribution, about what percent of the data lies within one, two, and three standard deviations of the mean?
Answer:
Mean is the average of the numbers.
Mean = Sum of observations/ no. of observations
68% of the normal distribution lies within one standard deviation of the mean.
95% of the normal distribution lies within two standard deviations of the mean.
99.7% of the normal distribution lies within three standard deviations of the mean.

Question 4.
Use the Internet or some other reference to find another data set that is normally distributed. Display your data in a histogram.
Answer:

A normal distribution has mean μ and standard deviation σ. Find the indicated probability for a randomly selected x-value from the distribution.

Question 1.
P(x ≤ μ)
Answer:
Given,
P(x ≤ μ)
Because of the symmetry of the normal curve, half of the values are on each side of the mean μ.
Thus, P(x ≤ μ)= 0.5

Question 2.
P(x ≥ μ)
Answer:
Given,
P(x ≥ μ)
Because of the symmetry of the normal curve, half of the values are on each side of the mean μ.
P(x ≥ μ) = 0.5

Question 3.
P(μ ≤ x ≤ μ + 2σ)
Answer:
Given,
P(μ ≤ x ≤ μ + 2σ)
The probability that a randomly selected x-value lies between μ and μ + 2σ is the marked area under the normal curve.
P(μ ≤ x ≤ μ + 2σ) = 0.34 + 0.135 = 0.475

Question 4.
P(μ − σ ≤ x ≤ μ)
Answer:
Given,
P(μ − σ ≤ x ≤ μ)
The total area under the related normal curve is 1
About 68% of the area lies within μ ± σ
About 95% of the area lies within μ ± 2σ
About 99.7% of the area lies within μ ± 3σ
compute the area under the curve bounded from the left with μ − σ and from the right with μ. thus, using the properties of the distribution
P(μ − σ ≤ x ≤ μ) = \(\frac { 0.68 }{ 2 } \) = 0.34
Thus P(μ − σ ≤ x ≤ μ) = 0.34

Question 5.
P(x ≤ μ − 3σ)
Answer:
Given,
P(x ≤ μ − 3σ)
The probability that the randomly selected x-value lies between μ and μ + 2σ is the marked area under the normal curve
P(x ≤ μ − 3σ) = 0.0015

Question 6.
P(x ≥ μ + σ)
Answer:
P(x ≥ μ + σ)
The probability that the randomly selected x-value lies between μ – σ and μ is the marked area under the normal curve

P(x ≥ μ + σ) = 0.135 + 0.0235 + 0.0015 = 0.16

Question 7.
WHAT IF?
In Example 2, about what percent of the people taking the test have scores between 43 and 79?
Answer:
Given,
The test has scored between 43 and 79
Let’s suppose that a normal distribution has a mean μ = 55 and a standard deviation σ = 12. Compute
P(43 ≤ x ≤ 79)
consider the following
μ – 43 = 55 – 43 = 12 = σ
25 = μ – σ
79 – μ = 79 – 55 = 24 = 2 . 12 = 2σ
7 = μ + 2σ
P(43 ≤ x ≤ 79) = P(μ – σ ≤ x ≤ μ + 2σ)
= P(μ – σ ≤ x ≤ μ) + P(μ ≤ x ≤ μ + 2σ)
= \(\frac { .68 }{ 2 } \) + \(\frac { 0.95 }{ 2 } \)
= 0.34 + 0.475
= 0.815
So, P(43 ≤ x ≤ 79) = 0.815

Question 8.
WHAT IF?
In Example 3, what is the probability that the infant weighs 3990 grams or more?
Answer:
If a normal distribution has a mean μ = 3270 and standard deviation σ = 600.
Compute
P(x ≥ 3990)
z = \(\frac { x – μ }{ σ } \)
P(x ≥ 3990) = P(\(\frac { x – 3270 }{ 600 } \) ≥\(\frac { 3990 – 3270 }{ 600 } \)) = P(z ≥ 1.2) = 1 – P(z<1.2)
P(z<1.2) = 0.8849
P(x ≥ 3990) = 1 – 0.8849 = 0.1151
The probability that the infant weighs 3990 grams or more is 0.1151

Question 9.
Explain why it makes sense that P(z ≤ 0) = 0.5.
Answer:
Given,
P(z ≤ 0) = 0.5
The area under the curve of the normal distribution is 1.
Thus, there are left of the mean will be half of it i.e 0.5. Compute this probability by using the z-values
z = \(\frac { x – μ }{ σ } \)
The z value for a particular x-value is called the z score for the x-value and is the number of standard deviations the x-value lies above or below the mean μ. The z-scores will be read directly from the table.
P(x≤μ) = P(\(\frac { x – μ }{ σ } \)<\(\frac { μ – μ }{ σ } \)) = P(z≤0)
P(x≤μ) = P(z≤0)
Since P(x≤μ) = 0.5
P(z ≤ 0) = 0.5

Question 10.
Determine whether the histogram has a normal distribution.

Answer:
The histogram is skewed left. So, the histogram does not have a normal distribution and you cannot use the normal distribution to interpret the histogram.

Using Normal Distributions 11.1 Exercises

Vocabulary and Core Concept Check
Question 1.
WRITING
Describe how to use the standard normal table to find P(z ≤ 1.4).
Answer:
Find the value where row 1 and column 4 intersect. The value is the probability is P(z ≤ 1.4).

Question 2.
WHICH ONE DOESN’T BELONG?
Which histogram does not belong with the other three? Explain your reasoning.

Answer:
The first diagram does not belong with the other three. Because the other diagrams are symmetric in the bell-shaped curve and the first diagram is not.

Monitoring Progress and Modeling with Mathematics

ATTENDING TO PRECISION In Exercises 3–6, give the percent of the area under the normal curve represented by the shaded region(s).
Question 3.

Answer:

Question 4.

Answer:
The total area under the related normal curve is 1
About 68% of the area lies within μ ± σ
About 95% of the area lies within μ ± 2σ
About 99.7% of the area lies within μ ± 3σ
Compute the area under the curve bounded from the left with μ − σ and from the right with μ. thus, using the properties of the distribution
\(\frac { (μ – 3σ) – (μ – σ) }{ 2 } \) = \(\frac { 99.7 – 68 }{ 2 } \) = 15.85%

Question 5.

Answer:

Question 6.

Answer:
The total area under the related normal curve is 1
About 68% of the area lies within μ ± σ
About 95% of the area lies within μ ± 2σ
About 99.7% of the area lies within μ ± 3σ
compute the area under the curve bounded from the left with μ − σ and from the right with μ. thus, using the properties of the distribution.
(μ – 2σ) – (μ – σ) = 95 – 68 = 27%

In Exercises 7–12, a normal distribution has mean μ and standard deviation σ. Find the indicated probability for a randomly selected x-value from the distribution.
Question 7.
P(x ≤ μ − σ)
Answer:

Question 8.
P(x ≥ μ − σ)
Answer:
The total area under the related normal curve is 1
About 68% of the area lies within μ ± σ
About 95% of the area lies within μ ± 2σ
About 99.7% of the area lies within μ ± 3σ
P(x ≥ μ − σ) = 1 – P(x < μ − σ)
= 1 – P(x ≤ μ − σ)
= 1 – 0.16
= 0.84
So, P(x ≥ μ − σ) = 0.84

Question 9.
P(x ≥ μ + 2σ)
Answer:

Question 10.
P(x ≤ μ + σ)
Answer:
The total area under the related normal curve is 1
About 68% of the area lies within μ ± σ
About 95% of the area lies within μ ± 2σ
About 99.7% of the area lies within μ ± 3σ
\(\frac { 1 + μ + σ }{ 2 } \) = \(\frac { 1 + 0.68 }{ 2 } \) = 0.84
So, P(x ≤ μ + σ) = 0.84

Question 11.
P(μ − σ ≤ x ≤ μ + σ)
Answer:

Question 12.
P(μ − 3σ ≤ x ≤ μ)
Answer:
The total area under the related normal curve is 1
About 68% of the area lies within μ ± σ
About 95% of the area lies within μ ± 2σ
About 99.7% of the area lies within μ ± 3σ
P(μ − 3σ ≤ x ≤ μ) = \(\frac { 0.997 }{ 2 } \)
= 0.4985
So, P(μ − 3σ ≤ x ≤ μ) = 0.4985

In Exercises 13–18, a normal distribution has a mean of 33 and a standard deviation of 4. Find the probability that a randomly selected x-value from the distribution is in the given interval.
Question 13.
between 29 and 37
Answer:

Question 14.
between 33 and 45
Answer:
Let the normal distribution has a mean μ = 33 and standard deviation σ = 4. Compute
P(33 ≤ x ≤ 45)
μ – 33 = 33 – 33 = 0
So, 33 = μ
45 – μ = 45 – 33 = 12 = 3 . 4 = 3σ
So, 45 = μ + 3σ
P(33 ≤ x ≤ 45) = P(3μ ≤ x ≤ μ + 3σ) = \(\frac { 0.997 }{ 2 } \) = 0.4985
P(33 ≤ x ≤ 45) = 0.4985

Question 15.
at least 25
Answer:

Question 16.
at least 29
Answer:
Let the normal distribution has a mean μ = 33 and standard deviation σ = 4. Compute
P(x ≥ 29)
μ – 29 = 33 – 29 = 4 = σ
So, 29 = μ – σ
P(x ≥ 29) = P(x ≥ μ – σ) = 0.5 + \(\frac { 0.68 }{ 2 } \) = 0.5 + 0.34
= 0.84
So, P(x ≥ 29) = 0.84

Question 17.
at most 37
Answer:

Question 18.
at most 21
Answer:
Let’s suppose that a normal distribution has a mean μ = 33 and standard deviation σ = 4. Compute
P is less than or equal to 21.
P(x ≤ 21)
μ – 21 = 33 – 21 = 12 = 3 . 4 = 3σ
So, 21 = μ + 3σ
P(x ≤ 21) = P(x ≤ μ – 3σ) = \(\frac { 1 – 0.997 }{ 2 } \) = 0.0015
P(x ≤ 21) = 0.0015

Question 19.
PROBLEM SOLVING
The wing lengths of houseflies are normally distributed with a mean of 4.6 millimeters and a standard deviation of 0.4 millimeter.

a. About what percent of houseflies have wing lengths between 3.8 millimeters and 5.0 millimeters?
b. About what percent of houseflies have wing lengths longer than 5.8 millimeters?
Answer:

Question 20.
PROBLEM SOLVING
The times a fire department takes to arrive at the scene of an emergency are normally distributed with a mean of 6 minutes and a standard deviation of 1 minute.
a. For about what percent of emergencies does the fire department arrive at the scene in 8 minutes or less?
Answer:
The time a fire department takes to arrive at the scene of an emergency is normally distributed with a mean of 6 minutes, σ = 1 min. Compute what percent of emergencies the fire department arrives at the scene in 8 minutes or less
P(x≤8)
8 – μ = 8 – 6 = 2 = 2 . 1 = 2σ
So, 8 = μ + 2σ
P(x≤8) = P(x≤μ + 2σ) = 0.5 + \(\frac { 0.95 }{ 2 } \)
= 0.5 + 0.475 = 0.975
About 97.5% fire department arrives at the scene in 8 minutes or less

b. The goal of the fire department is to reach the scene of an emergency in 5 minutes or less. About what percent of the time does the fire department achieve its goal?
Answer:
Compute what percent of the time the fire department achieves its goal
P((x≤5)
μ – 5 = 6 – 5 = 1 = σ
So, 5 = μ – σ
P((x≤5) = P((x≤μ – σ) = \(\frac {1 – 0.68 }{ 2 } \)
= 0.16
About 16% fire department reaches the goal.

ERROR ANALYSIS In Exercises 21 and 22, a normal distribution has a mean of 25 and a standard deviation of 2. Describe and correct the error in finding the probability that a randomly selected x-value is in the given interval.
Question 21.
between 23 and 27

Answer:

Question 22.
at least 21

Answer:
P(x ≥ 21) = P(x ≥ μ − 2σ) = 0.475 + 0.5 = 0.975

Question 23.
PROBLEM SOLVING
A busy time to visit a bank is during its Friday evening rush hours. For these hours, the waiting times at the drive-through window are normally distributed with a mean of 8 minutes and a standard deviation of 2 minutes. You have no more than 11 minutes to do your banking and still make it to your meeting on time. What is the probability that you will be late for the meeting?
Answer:

Question 24.
PROBLEM SOLVING
Scientists conducted aerial surveys of a seal sanctuary and recorded the number x of seals they observed during each survey. The numbers of seals observed were normally distributed with a mean of 73 seals and a standard deviation of 14.1 seals. Find the probability that at most 50 seals were observed during a randomly chosen survey.

Answer:
The number of seals observed were normally distributed with μ = 73 seals, σ = 14.1 seal
z = \(\frac { x – μ }{ σ } \)
P(x≤50) = P(\(\frac { x – 73 }{ 14.1 } \)≤\(\frac { 50 – 73 }{ 14.1 } \)) = P(z≤-1.6)
P(x≤50) = P(z≤-1.6) = 0.0548
The probability that at most 50 seals were observed during a randomly chosen survey is 0.0548

In Exercises 25 and 26, determine whether the histogram has a normal distribution.
Question 25.

Answer:

Question 26.

Answer:
The histogram is bell-shaped and fairly symmetric. Thus, the histogram has an approximately normal distribution.

Question 27.
ANALYZING RELATIONSHIPS
The table shows the numbers of tickets that are sold for various baseball games in a league over an entire season. Display the data in a histogram. Do the data fit a normal distribution? Explain.

Answer:

Question 28.
PROBLEM SOLVING
The guayule plant, which grows in the southwestern United States and in Mexico, is one of several plants that can be used as a source of rubber. In a large group of guayule plants, the heights of the plants are normally distributed with a mean of 12 inches and a standard deviation of 2 inches.

a. What percent of the plants are taller than 16 inches?
Answer:
The heights of the plants are normally distributed with μ = 12 in, σ = 2 in
z = \(\frac { x – μ }{ σ } \)
P(x≥16) = P(\(\frac { x – 12 }{ 2 } \)≥\(\frac { 16 – 12 }{ 2 } \)) = P(z≥2)
= 1 – P(z<2)
P(x≥16)= 1 – P(z<2) = 1 – 0.9772 = 0.0228
The percent of plants that are taller than 16 inches is about 2.28%

b. What percent of the plants are at most 13 inches?
Answer:
What percent of plants are at mos 13 inches = P(x≤13)
z = \(\frac { x – μ }{ σ } \)
P(x≤13) = P(\(\frac { x – 12 }{ 2 } \)≥\(\frac { 13 – 12 }{ 2 } \)) = P(z≤0.5)
P(x≤13) = P(z≤0.5) = 0.6915
The percent of plants that are at most 13 inches is about 69.15%

c. What percent of the plants are between 7 inches and 14 inches?
Answer:
What percent of plants are between 7 and 14 inches = P(7≤x≤14)
P(7≤x≤14) = P(\(\frac { 7 – 12 }{ 2 } \) ≤ \(\frac { x – 12 }{ 2 } \) ≤ \(\frac { 14 – 12 }{ 2 } \))
= P(-2.5≤z≤1)
= P(z≤1) – P(z≤-2.5)
= 0.8413 – 0.0062 = 0.8351
The percent of plants between 7 and 14 inches is about 83.51%

d. What percent of the plants are at least 3 inches taller than or at least 3 inches shorter than the mean height?
Answer:
What percent of plants are at least 3 inches taller than or at least 3 inches shorter than the mean weight
P(μ – 3 ≥ x ≥ μ + 3) = P(12 – 3 ≥ x ≥ 12 + 3)
= P(9 ≥ x ≥ 15)
= 1 – P(9≤x≤15)
P(9≤x≤15) = P(\(\frac { 9 – 12 }{ 2 } \) ≤ \(\frac { x – 12 }{ 2 } \) ≤ \(\frac { 15 – 12 }{ 2 } \))
= P(-1.5≤z≤1.5)
= P(z≤1.5) – P(z≤-1.5)
P(9 ≥ x ≥ 15) = 1 – P(z≤1.5) + P(z≤-1.5)
= 1 – 0.9332 + 0.0668
= 0.1336
The percent of plants that are at least 3 inches taller than or at least 3 inches shorter than the mean height is about 13.36%

Question 29.
REASONING
Boxes of cereal are filled by a machine. Tests show that the amount of cereal in each box varies. The weights are normally distributed with a mean of 20 ounces and a standard deviation of 0.25 ounce. Four boxes of cereal are randomly chosen.
a. What is the probability that all four boxes contain no more than 19.4 ounces of cereal?
b. Do you think the machine is functioning properly? Explain.
Answer:

Question 30.
THOUGHT PROVOKING
Sketch the graph of the standard normal distribution function, given by
f (x) = \(\frac{1}{\sqrt{2} \pi}\)e^−x2/2.
Estimate the area of the region bounded by the x-axis, the graph of f, and the vertical lines x =−3 and x = 3.
Answer:

P(-3≤z≤3) = P(z≤3) – P(z≤-3)
P(z≤3) = 0.9987
P(z≤-3) = 0.0013
P(-3≤z≤3) = P(z≤3) – P(z≤-3)
= 0.9987 – 0.0013 = 0.997

Question 31.
REASONING
For normally distributed data, describe the value that represents the 84th percentile in terms of the mean and standard deviation.
Answer:

Question 32.
HOW DO YOU SEE IT?
In the figure, the shaded region represents 47.5% of the area under a normal curve. What are the mean and standard deviation of the normal distribution?

Answer:
The shaded area represents 47.5% and since about 95% of the area lies within 2 standard deviations of the mean
Mean is the average of the numbers.
Mean = Sum of observations/ no. of observations
47.5% = \(\frac { 95% }{ 2 } \)
μ + 2σ= 16
μ + σ = 13
Subtract both equations
μ + 2σ – μ – σ = 16 – 13
σ = 3
Substitute σ = 3 in one equation
μ + 3 = 13
μ = 10

Question 33.
DRAWING CONCLUSIONS
You take both the SAT (Scholastic Aptitude Test) and the ACT (American College Test). You score 650 on the mathematics section of the SAT and 29 on the mathematics section of the ACT. The SAT test scores and the ACT test scores are each normally distributed. For the SAT, the mean is 514 and the standard deviation is 118. For the ACT, the mean is 21.0 and the standard deviation is 5.3.
a. What percentile is your SAT math score?
b. What percentile is your ACT math score?c. On which test did you perform better? Explain your reasoning.
Answer:

Question 34.
WRITING
Explain how you can convert ACT scores into corresponding SAT scores when you know the mean and standard deviation of each distribution.
Answer:
If we know the ACt z-score and change it into the z-formula for SAT test, we can obtain an equivalent score by solving the equation we get.
Let’s suppose that SAT and ACT test scores are normally distributed with mean μs, μa and standard deviations σs, σa. Let’s convert the ACT scores into corresponding SAT scores when you know the mean and SD of each distribution.
z-score of ACT is z<act>
P(xa < a) = P(\(\frac { xa – μa }{ σa } \) ≤ \(\frac { a – μa }{ σa } \)) = z<act>
P(xs < x) = z<act>
P(\(\frac { xs – μs }{ σs } \) ≤ \(\frac { X – μs }{ σs } \)) = z<act>
By solving the equation for x, we get the equivalent SAT score

Question 35.
MAKING AN ARGUMENT
A data set has a median of 80 and a mean of 90. Your friend claims that the distribution of the data is skewed left. Is your friend correct? Explain your reasoning.
Answer:

Question 36.
CRITICAL THINKING
The average scores on a statistics test are normally distributed with a mean of 75 and a standard deviation of 10. You randomly select a test score x. Find P (∣x−μ∣≥ 15 ).
Answer:
Let’s suppose that statistical tests are normally distributed with μ = 75, σ = 101
P(|x – μ| ≥ 15) = 1 – P(|x – μ| ≤ 15)
= 1 – P(-15 ≤ x – μ ≤ 15)
= 1 – P(μ – 15 ≤ x ≤ μ + 15)
= 1 – P(x≤ μ + 15) – P(x≤ μ – 15)
= 1 – P(x≤ 75 + 15) – P(x≤ 75 – 15)
= 1 – P(x≤90) + P(x≤60)
z = \(\frac { x – μ }{ σ } \)
P(x≤90) = P(\(\frac { x – 75 }{ 10 } \) ≤\(\frac { 90 – 75 }{ 10 } \))
= P(z≤1.5)
= 0.9332
P(x≤60) = P(\(\frac { x – 75 }{ 10 } \) ≤\(\frac { 60 – 75 }{ 10 } \))
= P(z≤-1.5)
= 0.0668
P(|x – μ| ≥ 15) = 1 – P(x≤90) + P(x≤60)
= 1 – 0.9332 + 0.0668
= 0.1336

Maintaining Mathematical Proficiency

Graph the function. Identify the x-intercepts and the points where the local maximums and local minimums occur. Determine the intervals for which the function is increasing or decreasing.
Question 37.
f(x) = x³ − 4x²+ 5
Answer:

Question 38.
g(x) = \(\frac{1}{4}\)x⁴ − 2x²−x − 3

Answer:

The x-intercepts are x = -2.8, x = 3.2
The local maximum value is -2.87, -5.07
The local minimum values are -1.86, -9.06
x∈(-∞, -1.86) U (-0.25, 2.11) function decreases
x ∈ (-1.86, -0.25) U(2.11, ∞) function increases.

Question 39.
h(x) = −0.5x²+ 3x + 7
Answer:

Question 40.
f(x) = −x⁴ + 6x² − 13
Answer:

There are no x-intercepts
The local maximum values are -4, 4
The local minimum values are -173, 1.73
x ∈ (-∞, -1.73) U (0, 1.73) function increases
x ∈ (-1.73, 0) U(1.73, ∞) function decreases

Lesson 11.2 Populations, Samples, and Hypotheses

Essential Question How can you test theoretical probability using sample data?

EXPLORATION 1

Using Sample Data
Work with a partner.
a. When two six-sided dice are rolled, what is the theoretical probability that you roll the same number on both dice?

b. Conduct an experiment to check your answer in part (a). What sample size did you use? Explain your reasoning.
c. Use the dice rolling simulator at BigIdeasMath.com to complete the table. Do your experimental data check the theoretical probability you found in part (a)? Explain. What happens as you increase the sample size?

EXPLORATION 2

Using Sample Data
Work with a partner.
a. When three six-sided dice are rolled, what is the theoretical probability that you roll the same number on all three dice?

b. Compare the theoretical probability you found in part (a) with the theoretical probability you found in Exploration 1(a).
c. Conduct an experiment to check your answer in part (a). How does adding a die affect the sample size that you use? Explain your reasoning.
d. Use the dice rolling simulator at BigIdeasMath.com to check your answer to part (a). What happens as you increase the sample size?

Communicate Your Answer

Question 3.
How can you test theoretical probability using sample data?
Answer:
Theoretical probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.
P(event) = number of favorable outcomes/total number of possible outcomes.
Sample Data:
A coin is tossed
P(head) = 1/2
P(tail) = 1/2.

Question 4.
Conduct an experiment to determine the probability of rolling a sum of 7 when two six-sided dice are rolled. Then find the theoretical probability and compare your answers.
Answer:
Total number of outcomes = 6 × 6 = 36
Favorable outcomes = (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)
P(A) = 6/36 = 1/6
So, the probability of getting a sum as 7 when two dice are thrown is 1/6.

Monitoring Progress

In Monitoring Progress Questions 1 and 2, identify the population and the sample.
Question 1.
To estimate the retail prices for three grades of gasoline sold in the United States, the Energy Information Association calls 800 retail gasoline outlets, records the prices, and then determines the average price for each grade.
Answer:
In the population of all retail gasoline outlets, a sample of 800 retail gasoline outlets is chosen to participate in the survey.

Question 2.
A survey of 4464 shoppers in the United States found that they spent an average of $407.02 from Thursday through Sunday during a recent Thanksgiving holiday.
Answer:
In the population of all shoppers in the United States, a sample of 4464 shoppers are chosen to participate in the survey.

Question 3.
A survey found that the median salary of 1068 statisticians is about $72,800. Is the median salary a parameter or a statistic? Explain your reasoning.
Answer:
Since the median salary is calculated for a sample of 1068 statisticians, not the entire population, this median salary cannot represent the whole population. Thus, it is a statistic.

Question 4.
The mean age of U.S. representatives at the start of the 113th Congress was about 57 years. Is the mean age a parameter or a statistic? Explain your reasoning.
Answer:
Since the average age is computed for all representatives, this average represents a parameter.

Question 5.
WHAT IF?
In Example 3, what should you conclude when you roll the actual die 50 times and get (a) 24 odd numbers and (b) 31 odd numbers?
Answer:
a. Getting 24 odd numbers in 50 rolls corresponds to a proportion of \(\frac { 24 }{ 50 } \) = 0.48. In the simulation, this result has a relative frequency of 0.12. Most of the results are close to 0.50. Because this result can easily occur by chance, you can conclude that the maker’s claim is most likely true.

b. Getting 35 odd numbers in 50 rolls corresponds to a proportion of \(\frac { 31 }{ 50 } \) = 0.62. In the simulation, this result occurs with a small frequency which is about 0.02. Because getting 31 odd numbers is highly unlikely to occur by chance, you can conclude that the maker’s claim is most likely false.

Populations, Samples, and Hypotheses 11.2 Exercises

Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
A portion of a population that can be studied in order to make predictions about the entire population is a(n) ___________.
Answer:
A portion of a population that can be studied in order to make predictions about the entire population is a sample.

Question 2.
WRITING
Describe the difference between a parameter and a statistic. Give an example of each.
Answer:
Parameters are numbers that describe data for an entire population. Example: population mean.
Statistics are numbers that describe data from a sample i.e some subset of the entire population. Example: Sample mean.

Question 3.
VOCABULARY
What is a hypothesis in statistics?
Answer:
The hypothesis is a type of statistical analysis in which you put your assumptions about a parameter to the test. A hypothesis in statistics is a claim about a characteristic of a population.

Question 4.
WRITING
Describe two ways you can make an incorrect decision when analyzing a hypothesis.
Answer:
A type I error occurs if an investigator rejects a null hypothesis that is actually true in the population.
A type II error occurs if the investigator fails to reject a null hypothesis that is actually false in the population.

Monitoring Progress and Modeling with Mathematics

In Exercises 5–8, determine whether the data are collected from a population or a sample. Explain your reasoning.
Question 5.
the number of high school students in the United States
Answer:
It is a population mean because data are collected from a population because every high school is counted.

Question 6.
the color of every third car that passes your house
Answer:
It is a sample mean because it was counted as a subset of the entire population.

Question 7.
a survey of 100 spectators at a sporting event with 1800 spectators
Answer:

Question 8.
the age of each dentist in the United States
Answer:
It is a population because it is counted as the age of each dentist in the United States.

In Exercises 9–12, identify the population and sample. Describe the sample.
Question 9.
In the United States, a survey of 1152 adults ages 18 and over found that 403 of them pretend to use their smartphones to avoid talking to someone.

Answer:

Question 10.
In the United States, a survey of 1777 adults ages 18 and over found that 1279 of them do some kind of spring cleaning every year.
Answer:
Given,
In the population of all adults ages 18 and over in the United States, a sample of 1777 adults is surveyed.
In the survey, 1279 of them do some kind of spring cleaning every year and
1777 – 1279 = 489 do not.

Question 11.
In a school district, a survey of 1300 high school students found that 1001 of them like the new, healthy cafeteria food choices.
Answer:

Question 12.
In the United States, a survey of 2000 households with at least one child found that 1280 of them eat dinner together every night.

Answer:
Given,
In the United States, a survey of 2000 households with at least one child found that 1280 of them eat dinner together every night.
In the population of all households with at least one child in the United States, a sample of 2000 households is surveyed.

In Exercises 13–16, determine whether the numerical value is a parameter or a statistic. Explain your reasoning.
Question 13.
The average annual salary of some physical therapists in a state is $76,210.

Answer:

Question 14.
In a recent year, 53% of the senators in the United States Senate were Democrats.

Answer:
The numerical value is a parameter. Since we have that 47 percent of all senators in the United States senate were Democrats, it represents a parameter.

Question 15.
Seventy-three percent of all the students in a school would prefer to have school dances on Saturday.
Answer:

Question 16.
A survey of U.S. adults found that 10% believe a cleaning product they use is not safe for the environment.
Answer:
Given,
A survey of U.S. adults found that 10% believe a cleaning product they use is not safe for the environment.
Since we have a survey where it’s found that 10% of some adults, not the entire population, believe that a cleaning product they use is not safe for the environment. Hence it is a statistic.

Question 17.
ERROR ANALYSIS
A survey of 1270 high school students found that 965 students felt added stress because of their workload. Describe and correct the error in identifying the population and the sample.

Answer:

Question 18.
ERROR ANALYSIS
Of all the players on a National Football League team, the mean age is 26 years. Describe and correct the error in determining whether the mean age represents a parameter or statistic. Because the mean age of 26 is based only on one football team, it is a statistic.

Answer:
It is not a statistic parameter. Since we have that the mean age of all the players on a National Football League team is 26.

Question 19.
MODELING WITH MATHEMATICS
You flip a coin 4 times and do not get a tails. You suspect this coin favors heads. The coin maker claims that the coin doesnot favor heads or tails. You simulate flipping the coin 50 times by repeatedly drawing 200 random samples of size 50. The histogram shows the results. What should you conclude when you flip the actual coin 50 times and get (a) 27 heads and (b) 33 heads?

Answer:

Question 20.
MODELING WITH MATHEMATICS
Use the histogram in Exercise 19 to determine what you should conclude when you flip the actual coin 50 times and get (a) 17 heads and (b) 23 heads.

Answer:
a. Let’s suppose that we flip the actual coin 50 times and get 17 heads.
Getting 17 heads in 50 flipping corresponds to a proportion of \(\frac { 17 }{ 50 } \) = 0.34.
In the simulation, this result occurs with a small frequency.
Because getting 17 heads is highly unlikely to occur by chance, you can conclude that the maker’s claim is most likely false.

b. Let’s suppose that we are flip the actual coin 50 times and get 23 heads.
Getting 23 heads in 50 flipping corresponds to a proportion of \(\frac { 23 }{ 50 } \) = 0.46.
In the simulation, this result had a relative frequency of 0.10. In fact, most of the results are close to 0.50.
Because getting this result can easily occur by chance, you can conclude that the makers claim is most likely true.

Question 21.
MAKING AN ARGUMENT
A random sample of five people at a movie theater from a population of 200 people gave the film 4 out of 4 stars. Your friend concludes that everyone in the movie theater would give the film 4 stars. Is your friend correct? Explain your reasoning.
Answer:

Question 22.
HOW DO YOU SEE IT?
Use the Venn diagram to identify the population and sample. Explain your reasoning.

Answer:
A population will be a major of students at a university and the sample is the majors of students at a university that takes chemistry.

Question 23.
OPEN-ENDED
Find a newspaper or magazine article that describes a survey. Identify the population and sample. Describe the sample.
Answer:

Question 24.
THOUGHT PROVOKING
You choose a random sample of 200 from a population of 2000. Each person in the sample is asked how many hours of sleep he or she gets each night. The mean of your sample is 8 hours. Is it possible that the mean of the entire population is only 7.5 hours of sleep each night? Explain.
Answer:
Given,
You choose a random sample of 200 from a population of 2000.
Mean = 8 hours
Yes, it is possible because the mean of your sample is 8 hours, which is very close to 7.5 hours. Also, the random sample of 200 people is enough big size of the sample to estimate the mean of the population.

Question 25.
DRAWING CONCLUSIONS
You perform two simulations of repeatedly selecting a marble out of a bag with replacement that contains three red marbles and three blue marbles. The first simulation uses 20 random samples of size 10, and the second uses 400 random samples of size 10. The histograms show the results. Which simulation should you use to accurately analyze a hypothesis? Explain.

Answer:

Question 26.
PROBLEM SOLVING
You roll an eight-sided die five times and get a four every time. You suspect that the die favors the number four. The die maker claims that the die does not favor any number.
a. Perform a simulation involving 50 trials of rolling the actual die and getting a four to test the die maker’s claim. Display the results in a histogram.
b. What should you conclude when you roll the actual die 50 times and get 20 fours? 7 fours?

Answer:

Let’s suppose that we are rolling the eight-side die 50 times and get 20 fours. Getting 20 fours in 50 rollings corresponds to a proportion of \(\frac { 20 }{ 50 } \) = 0.4. In the simulation, this result does not occur. Because getting 20 fours is highly unlikely to occur by chance, you can conclude that the maker’s claim is most likely false.
Let’s suppose that we are rolling the eight side die 50 times and get 7 fours. Getting 7 fours in 50 rollings corresponds to a proportion of \(\frac { 7 }{ 50 } \) = 0.14.
In the simulation, this result had a relative frequency of 0.16. Because this result can easily occur by chance, you can conclude that the maker’s claim is most likely true.

Maintaining Mathematical Proficiency

Solve the equation by completing the square.
Question 27.
x²− 10x − 4 = 0
Answer:

Question 28.
3t² + 6t = 18
Answer:
Given,
3t² + 6t = 18
3t² + 6t – 180 = 0
3(t² + 2t) – 18 = 0
3(t² + 2t) – 18 + 3 – 3 = 0
3(t² + 2t + 1) – 21 = 0
3(t + 1)² – 21 = 0
3(t + 1)² = 21
(t + 1)² = 7
t + 1 = ±√7
t = ±√7 – 1
The solutions are -1 + √7, √7 + 1

Question 29.
s² + 10s + 8 = 0
Answer:

Solve the equation using the Quadratic Formula.
Question 30.
n²+ 2n + 2 = 0
Answer:
Given,
n²+ 2n + 2 = 0
n² + 2n + 1 + 1 = 0
(n + 1)² + 1 = 0
(n + 1)²= -1
n + 1 = ±i
n = ±i – 1
The solutions are -1 + i, -1 – i

Question 31.
4z² + 28z = 15
Answer:

Question 32.
5w − w² = −11
Answer:
Given,
w = \(\frac { -5 ± √(5² – 4(-1)(11)) }{ 2(-1) } \)
= \(\frac { -5 ± √(25 + 44) }{ -2 } \)
= \(\frac { -5 ± √69 }{ -2 } \)
The solutions of the equations are \(\frac { -5 + √69 }{ -2 } \), \(\frac { -5 – √69 }{ -2 } \)
The solutions of the equations are \(\frac { -5 + √69 }{ -2 } \), \(\frac { -5 – √69 }{ -2 } \)

Lesson 11.3 Collecting Data

Essential Question What are some considerations when undertaking a statistical study?
Answer:
The objective of the statistical study is to collect the data and then utilize the data to make a decision. Any decision you make utilizing the results of a statistical study is essential as the process used to get the data. If the process is flawed, then the resulting decision is problematic.

EXPLORATION 1

Analyzing Sampling Techniques
Work with a partner. Determine whether each sample is representative of the population. Explain your reasoning.

a. To determine the number of hours people exercise during a week, researchers use random-digit dialing and call 1500 people.
b. To determine how many text messages high school students send in a week, researchers post a survey on a website and receive 750 responses.
c. To determine how much money college students spend on clothes each semester,a researcher surveys 450 college students as they leave the university library.
d. To determine the quality of service customers receive, an airline sends an e-mail survey to each customer after the completion of a flight.

EXPLORATION 2

Analyzing Survey Questions
Work with a partner. Determine whether each survey question is biased. Explain your reasoning. If so, suggest an unbiased rewording of the question.
a. Does eating nutritious, whole-grain foods improve your health?
b. Do you ever attempt the dangerous activity of texting while driving?
c. How many hours do you sleep each night?
d. How can the mayor of your city improve his or her public image?

EXPLORATION 3

Analyzing Survey Randomness and Truthfulness
Work with a partner. Discuss each potential problem in obtaining a random survey of a population. Include suggestions for overcoming the problem.
a. The people selected might not be a random sample of the population.
b. The people selected might not be willing to participate in the survey.
c. The people selected might not be truthful when answering the question.
d. The people selected might not understand the survey question.

Communicate Your Answer

Question 4.
What are some considerations when undertaking a statistical study?
Answer:
The 4 considerations when undertaking a statistical study are as follows
1. what is the research question?
2. What is the study design?
3. What is the scale of measurement?
4. Any data issues?

Question 5.
Find a real-life example of a biased survey question. Then suggest an unbiased rewording of the question.
Answer:

Monitoring Progress

Question 1.
WHAT IF?
In Example 1, you divide the students in your school according to their zip codes, then select all of the students that live in one zip code. What type of sample are you using?
Answer:
Divide the students in our school according to their zip codes, then select all of the students that live in one zip code. Thus, we use a cluster-type sample.

Question 2.
Describe another method you can use to obtain a stratified sample in Example 1.
Answer:
Divide subjects into subgroups called strata based on the characteristics they share. For example gender, and educational accomplishment. Once divided, then each subgroup is randomly sampled using another probability sampling method.

Question 3.
The manager of a concert hall wants to know how often people in the community attend concerts. The manager asks 45 people standing in line for a rock concert how many concerts they attend per year. Identify the type of sample the manager is using and explain why the sample is biased.
Answer:
Given,
The manager of a concert hall wants to know how often people in the community attend concerts.
The manager asks 45 people which stand in line for a rock concert it is a convenience sample because only members of a population who are easy to reach are asked.

Question 4.
In Example 3, what is another method you can use to generate a random sample of 50 students? Explain why your sampling method is random.
Answer:
You have 246 seniors and you have to randomly select 50 students.
We can write the names of all the students on paper and put them in a hat.
Take 50 papers out of the hat and form a pattern.
It’s a random sample because we don’t know which piece of paper we are going to pull out of the hat.

Identify the method of data collection the situation describes.
Question 5.
Members of a student council at your school ask every eighth student who enters the cafeteria whether they like the snacks in the school’s vending machines.
Answer:
Given that,
Members of a student council at your school ask every eighth student who enters the cafeteria whether they like the snacks in the school’s vending machines.
After every eighth student who enters the cafeteria is asked for an opinion, we have a survey.

Question 6.
A park ranger measures and records the heights of trees in a park as they grow.
Answer:
Given that,
A park ranger measures and records the heights of trees in a park as they grow.
We have a park ranger that measures and record the heights of trees in a park as they grow, and we have an observational study.

Question 7.
A researcher uses a computer program to help determine how fast an influenza virus might spread within a city.
Answer:
Given that,
A researcher uses a computer program to help determine how fast an influenza virus might spread within a city.
We have that researcher collecting data by using the computer, we have a simulation.

Question 8.
Explain why the survey question below may be biased or otherwise introduce bias into the survey. Then describe a way to correct the flaw.
“Do you agree that our school cafeteria should switch to a healthier menu?”
Answer:
The question is biased because it encourages the answer yes.

Collecting Data 11.3 Exercises

Vocabulary and Core Concept Check
Question 1.
VOCABULARY
Describe the difference between a stratified sample and a cluster sample.
Answer:
It is a stratified sample, after the groups are formed, a random sample is selected from each group are formed, and all the members of one or more groups are randomly selected.

Question 2.
COMPLETE THE SENTENCE
A sample for which each member of a population has an equal chance of being selected is a(n) __________ sample.
Answer:
A sample for which each member of a population has an equal chance of being selected is random sample.

Question 3.
WRITING
Describe a situation in which you would use a simulation to collect data.
Answer: You would use a simulation to collect data to determine how quickly an oil spill would spread through a lake.

Question 4.
WRITING
Describe the difference between an unbiased sample and a biased sample. Give one example of each.
Answer: In a biased sample, one or more parts of the population are favored over others, whereas in an unbiased sample, each member of the population has an equal chance of being selected. An unbiased statistic sample would be expected to have a difference of 0 over time.

Monitoring Progress and Modeling with Mathematics

In Exercises 5–8, identify the type of sample described.
Question 5.
The owners of a chain of 260 retail stores want to assess employee job satisfaction. Employees from 12 stores near the headquarters are surveyed.
Answer:

Question 6.
Each employee in a company writes their name on a card and places it in a hat. The employees whose names are on the first two cards drawn each win a gift card.
Answer:
Given that,
Each employee in a company writes their name on a card and places it in a hat. The employees whose names are on the first two cards drawn each win a gift card.
Since each of the employees in a company has the same chance to be drawn the samples is a random.

Question 7.
A taxicab company wants to know whether its customers are satisfied with the service. Drivers survey every tenth customer during the day.
Answer:

Question 8.
The owner of a community pool wants to ask patrons whether they think the water should be colder. Patrons are divided into four age groups, and a sample is randomly surveyed from each age group.
Answer:
Given that,
The owner of a community pool wants to ask patrons whether they think the water should be colder. Patrons are divided into four age groups, and a sample is randomly surveyed from each age group.
You are dividing the patrons into four age groups. So the sample is a stratified sample.

In Exercises 9–12, identify the type of sample and explain why the sample is biased.
Question 9.
A town council wants to know whether residents support having an off-leash area for dogs in the town park. Eighty dog owners are surveyed at the park.

Answer:

Question 10.
A sportswriter wants to determine whether baseball coaches think wooden bats should be mandatory in collegiate baseball. The sportswriter mails surveys to all collegiate coaches and uses the surveys that are returned.
Answer:
Given data, A sportswriter wants to determine whether baseball coaches think wooden bats should be mandatory in collegiate baseball. The sportswriter mails surveys to all collegiate coaches and uses the surveys that are returned.
The participants in the survey are collegiate coaches and using the surveys that are returned, it is a self-selected sample.

Question 11.
You want to find out whether booth holders at a convention were pleased with their booth locations. You divide the convention center into six sections and survey every booth holder in the fifth section.
Answer:

Question 12.
Every tenth employee who arrives at a company health fair answers a survey that asks for opinions about new health-related programs.
Answer:
Given that,
Every tenth employee who arrives at a company health fair answers a survey that asks for opinions about new health-related programs.
Since every tenth employee who arrives at a company health fair answers a survey, it is a systematic sample.

Question 13.
ERROR ANALYSIS
Surveys are mailed to every other household in a neighborhood. Each survey that is returned is used. Describe and correct the error in identifying the type of sample that is used.

Answer:

Question 14.
ERROR ANALYSIS
A researcher wants to know whether the U.S. workforce supports raising the minimum wage. Fifty high school students chosen at random are surveyed. Describe and correct the error in determining whether the sample is biased.

Answer:
Given data A researcher wants to know whether the U.S. workforce supports raising the minimum wage. Fifty high school students chosen at random are surveyed.
Since the sample is not chosen from the entire population it cannot be a non-biased sample. Thus, a sample is biased.

In Exercises 15–18, determine whether the sample is biased. Explain your reasoning.
Question 15.
Every third person who enters an athletic event is asked whether he or she supports the use of instant replay in officiating the event.
Answer:

Question 16.
A governor wants to know whether voters in the state support building a highway that will pass through a state forest. Business owners in a town near the proposed highway are randomly surveyed.
Answer:
Participants in the survey are only random business owners in a town near the proposed highway and they cannot represent the whole population of the town.
It is a biased sample.

Question 17.
To assess customers’ experiences making purchases online, a rating company e-mails purchasers and asks that they click on a link and complete a survey.
Answer:

Question 18.
Your school principal randomly selects five students from each grade to complete a survey about classroom participation.
Answer:
Given data, Your school principal randomly selects five students from each grade to complete a survey about classroom participation.
Since we chose 5 students from each grade to complete a survey about classroom participation, the sample represents the whole population.
Thus, the sample is not biased.

Question 19.
WRITING
The staff of a student newsletter wants to conduct a survey of the students’ favorite television shows. There are 1225 students in the school. Describe a method for selecting a random sample of 250 students to survey.

Answer:

Question 20.
WRITING
A national collegiate athletic association wants to survey 15 of the 120 head football coaches in a division about a proposed rules change. Describe a method for selecting a random sample of coaches to survey.
Answer:
Given,
A national collegiate athletic association wants to survey 15 of the 120 head football coaches in a division about a proposed rules change.
We can divide 120 coaches into three groups with 40 coaches, and from each group choose 5 coaches.
For example, we can denote each coach from the group by some integer number from 1 to 40, and by using the random integer generator chooses 5 coaches.
In the described way, from all 3 groups, we can choose 15 coaches.

In Exercises 21–24, identify the method of data collection the situation describes.
Question 21.
A researcher uses technology to estimate the damage that will be done if a volcano erupts.

Answer:

Question 22.
The owner of a restaurant asks 20 customers whether they are satisfied with the quality of their meals.
Answer:
The owner of a restaurant asks 20 customers whether they are satisfied with the quality of their meals which is an investigation of one or more characteristics of a population. So, this situation is a survey.

Question 23.
A researcher compares incomes of people who live in rural areas with those who live in large urban areas.
Answer:

Question 24.
A researcher places bacteria samples in two different climates. The researcher then measures the bacteria growth in each sample after 3 days.
Answer:
Given data, A researcher places bacteria samples in two different climates. The researcher then measures the bacteria growth in each sample after 3 days.
We have a treatment on individuals in order to collect data on their responses to the treatment. The situation describes an experiment.

In Exercises 25–28, explain why the survey question may be biased or otherwise introduce bias into the survey. Then describe a way to correct the flaw.
Question 25.
“Do you agree that the budget of our city should be cut?”
Answer:

Question 26.
“Would you rather watch the latest award-winning movie or just read some book?”
Answer:
“Would you rather watch the latest award-winning movie or just read some book?”
The question favors the answer that you rather watch the movie.

Question 27.
“The tap water coming from our western water supply contains twice the level of arsenic of water from our eastern supply. Do you think the government should address this health problem?”
Answer:

Question 28.
A child asks, “Do you support the construction of a new children’s hospital?”
Answer:
A child asks, “Do you support the construction of a new children’s hospital?” The question favors the answer yes.

In Exercises 29–32, determine whether the survey question may be biased or otherwise introduce bias into the survey. Explain your reasoning.
Question 29.
“Do you favor government funding to help prevent acid rain?”
Answer:

Question 30.
“Do you think that renovating the old town hall would be a mistake?”
Answer:
“Do you think that renovating the old town hall would be a mistake?”
The question is biased because it encourages the answer yes.

Question 31.
A police officer asks mall visitors, “Do you wear your seat belt regularly?”
Answer:

Question 32.
“Do you agree with the amendments to the Clean Air Act?”
Answer:
The question “Do you agree with the amendments to the Clean Air Act?” is not biased because it is asked without much information in it so that the right opinion of the survey participants can be obtained.

Question 33.
REASONING
A researcher studies the effect of fiber supplements on heart disease. The researcher identified 175 people who take fiber supplements and 175 people who do not take fiber supplements. The study found that those who took the supplements had 19.6% fewer heart attacks. The researcher concludes that taking fiber supplements reduces the chance of heart attacks.

a. Explain why the researcher’s conclusion may not be valid.
b. Describe how the researcher could have conducted the study differently to produce valid results.
Answer:

Question 34.
HOW DO YOU SEE IT?
A poll is conducted to predict the results of a statewide election in New Mexico before all the votes are counted. Fifty voters in each of the state’s 33 counties are asked how they voted as they leave the polls.
a. Identify the type of sample described.
b. Explain how the diagram shows that the polling method could result in a biased sample.

Answer:
a. The situation describes the Stratified sample.
b. With the date of the damp, we can notice that different countries have different populations. The sample consists of an equal number of respondents from each country and the number of respondents from the country is not proportional to the size of the country. Therefore, a small sample from a country with a large population cannot be considered credible.

Question 35.
WRITING
Consider each type of sample listed on page 610. Which of the samples are most likely to lead to biased results? Explain.
Answer:

Question 36.
THOUGHT PROVOKING
What is the difference between a “blind experiment” and a “double-blind experiment?” Describe a possible advantage of the second type of experiment over the first.
Answer:
In a blind or blind experiment, information that may influence the participants of the experiment is withheld until after the experiment is complete.
A double-blind experiment is one in which neither the participant nor the experimenters know who is receiving a particular treatment. this procedure is utilized to prevent bias in research results.
Double-blind experiments are particularly useful for preventing bias due to demand characteristics or the placebo effect which represent a possible advantage of double-blind experiments.

Question 37.
WRITING
A college wants to survey its graduating seniors to find out how many have already found jobs in their field of study after graduation.
a. What is the objective of the survey?
b. Describe the population for the survey.
c. Write two unbiased questions for the survey.
Answer:

Question 38.
REASONING
About 3.2% of U.S. adults follow a vegetarian-based diet. Two randomly selected groups of people were asked whether they follow such a diet. The first sample consists of 20 people and the second sample consists of 200 people. Which sample proportion is more likely to be representative of the national percentage? Explain.
Answer:
Given data is
About 3.2% of U.S. adults follow a vegetarian-based diet. Two randomly selected groups of people were asked whether they follow such a diet. The first sample consists of 20 people and the second sample consists of 200 people.
The larger sample (second sample) will better represent the whole population because some unusual responses will not affect the results much as with the smaller sample (first sample).

Question 39.
MAKING AN ARGUMENT
The U.S. Census is taken every10 years to gather data from the population. Your friend claims that the sample cannot be biased. Is your friend correct? Explain.
Answer:

Question 40.
OPEN-ENDED
An airline wants to know whether travelers have enough leg room on its planes.

a. What method of data collection is appropriate for this situation?
b. Describe a sampling method that is likely to give biased results. Explain.
c. Describe a sampling method that is not likely to give biased results. Explain.
d. Write one biased question and one unbiased question for this situation.
Answer:
a. An airline wants to know whether travelers have enough legroom on its planes. The situation describes the survey method.
b. We can conduct the survey by mail. Only people with strong opinions about the legroom on its planers will take part in that survey and return the survey by mail.
c. For example, in order to obtain a non-biased survey, an airline may grant a certain discount on tickets if it participates in the survey in all class seats. That way more people will take part in the survey and we will get an unbiased sample.
d. Biased question: “Do you agree that the legroom is too small?”
Non-biased question: “Do you have enough legroom?”

Question 41.
REASONING
A website contains a link to a survey that asks how much time each person spends on the Internet each week.
a. What type of sampling method is used in this situation?
b. Which population is likely to respond to the survey? What can you conclude?
Answer:

Maintaining Mathematical Proficiency

Evaluate the expression without using a calculator.
Question 42.
4^5/2

Answer:
Given,
4^5/2 = (2²)^{\(\frac { 5 }{ 2 } \)}
= 2 ^{2 . \(\frac { 5 }{ 2 } \)}
= 2⁵
= 32
So, 4^5/2 = 32

Question 43.
27^2/3
Answer:

Question 44.
−64^1/3

Answer:
Given,
−64^1/3 = -((4)³)^1/3
= -4^{3 . 1/3}
= -4
So, −64^1/3 is -4.

Question 45.
8^-2/3
Answer:

Simplify the expression.
Question 46.
(4^3/2 • 4^1/4)⁴

Answer:
(4^3/2 • 4^1/4)⁴ = (4^{3/2 + 1/4})⁴
=(4^7/4>/sup>)4
=4^{7/4 . 4}
= 4⁷
= 16384

Question 47.
(6^1/3 • 3^1/3)^-2
Answer:

Question 48.
\(\sqrt [ 3 ]{ 4 }\) • \(\sqrt [ 3 ]{ 16 }\)
Answer:
\(\sqrt [ 3 ]{ 4 }\) • \(\sqrt [ 3 ]{ 16 }\) = \(\sqrt [ 3 ]{ 4 }\) • \(\sqrt [ 3 ]{ 4² }\)
= 4^1/3 + 4^2/3
= 4^{1/3 + 2/3}
= 4
So, \(\sqrt [ 3 ]{ 4 }\) • \(\sqrt [ 3 ]{ 16 }\) = 4

Question 49.
\(\frac{\sqrt[4]{405}}{\sqrt[4]{5}}\)
Answer:

Data Analysis and Statistics Study Skills: Reworking Your Notes

11.1–11.3 What Did You Learn?

Core Vocabulary

Core Concepts

Mathematical Practices

Question 1.
What previously established results, if any, did you use to solve Exercise 31 on page 602?
Answer:

Question 2.
What external resources, if any, did you use to answer Exercise 36 on page 616?
Answer:

Study Skills: Reworking Your Notes

It’s almost impossible to write down in your notes all the detailed information you are taught in class. A good way to reinforce the concepts and put them into your long-term memory is to rework your notes. When you take notes, leave extra space on the pages. You can go back after class and fill in:

• important definitions and rules
• additional examples
• questions you have about the material
  

Data Analysis and Statistics 11.1–11.3 Quiz

A normal distribution has a mean of 32 and a standard deviation of 4. Find the probability that a randomly selected x-value from the distribution is in the given interval.

Question 1.
at least 28
Answer:
Given,
A normal distribution has a mean of 32 and a standard deviation of 4
Mean μ = 32
σ = 4
We have to determine P(x > 28)
28 is one standard deviation to the left of the mean

P(x > 28) = P(x > μ – σ)
= 34% + 34% + 13.5% + 2.35% + 0.015%
= 84%
So, P(x > 28) = 84%

Question 2.
between 20 and 32
Answer:
A normal distribution has a mean of 32 and a standard deviation of 4
Mean μ = 32
σ = 4
μ – 20 = 32 – 20 = 12 = 3 . 4 = 3σ
20 = μ – 3σ
32 – μ = 32 – 32 = 0
32 = μ
P(20≤x≤32) = P(μ – 3σ≤x≤μ)
= P(x≤μ) – P(x≤μ – 3σ)
= 0.5 – 0.0015
= 0.4985
The probability is 0.4985

Question 3.
at most 26
Answer:
A normal distribution has a mean of 32 and a standard deviation of 4
Mean μ = 32
σ = 4
Now compute P(x≤26)
z = \(\frac { x – μ }{ σ } \)
P(x≤26) = P(\(\frac { x – 32 }{ 4 } \)≤ \(\frac { 26 – 32 }{ 4 } \))
= P(z≤-1.5)
= 0.0668
The probability is 0.0668. (6.68%)

Question 4.
at most 35
Answer:
A normal distribution has a mean of 32 and a standard deviation of 4
Mean μ = 32
σ = 4
Now compute P(x≤35)
z = \(\frac { x – μ }{ σ } \)
P(x≤35) = P(\(\frac { x – 32 }{ 4 } \)≤ \(\frac { 35 – 32 }{ 4 } \))
= P(z≤0.8)
= 0.7881
The probability is 0.7881 (78.8%)

Determine whether the histogram has a normal distribution.
Question 5.

Answer:
The variable that is normally distributed has a histogram that is bell-shaped with only one peak and symmetric around the mean.
The above graph has one peak so we can say that the histogram has a normal distribution.

Question 6.

Answer:
The variable that is normally distributed has a histogram that is bell-shaped with only one peak and symmetric around the mean.
The above graph does not have a normal distribution.

Question 7.
A survey of 1654 high school seniors determined that 1125 plan to attend college. Identify the population and the sample. Describe the sample.
Answer:
Given,
A survey of 1654 high school seniors determined that 1125 plan to attend college. Identify the population and the sample.
Sample: The 1654 high school seniors surveyed. The sample consists of 1125 high school seniors who plan to attend college and 529 high school seniors who do not.
Population: Al high school students

Question 8.
A survey of all employees at a company found that the mean one-way daily commute to work of the employees is 25.5 minutes. Is the mean time a parameter or a statistic? Explain your reasoning.
Answer:
Given,
A survey of all employees at a company found that the mean one-way daily commute to work of the employees is 25.5 minutes.
The mean time is a parameter.
The entire population was surveyed.

Question 9.
A researcher records the number of bacteria present in several samples in a laboratory. Identify the method of data collection.
Answer:
Given,
A researcher records the number of bacteria present in several samples in a laboratory. The records of samples in the lab will be observational.
The method of data collection is an Observational study.

Question 10.
You spin a five-color spinner, which is divided into equal parts, five times and every time the spinner lands on red. You suspect the spinner favors red. The maker of the spinner claims that the spinner does not favor any color. You simulate spinning the spinner 50 times by repeatedly drawing 200 random samples of size 50. The histogram shows the results. Use the histogram to determine what you should conclude when you spin the actual spinner 50 times and the spinner lands on red (a) 9 times and (b) 19 times.

Answer:
a) The marker’s claim is most likely true
b) The marker’s claim is most likely false

Question 11.
A local television station wants to find the number of hours per week people in the viewing area watch sporting events on television. The station surveys people at a nearby sports stadium.
a. Identify the type of sample described.
Answer: The type of sample is Convenience sample

b. Is the sample biased? Explain your reasoning.
Answer: Yes, people at a sports stadium are more likely to watch more sports on television than people who never go to a sports stadium.

c. Describe a method for selecting a random sample of 200 people to survey.
Answer:
Randomly pick 200 members from the seating area by placing all seat numbers in a hat.

Lesson 11.4 Experimental Design

Essential Question How can you use an experiment to test a conjecture?

EXPLORATION 1

Using an Experiment
Work with a partner. Standard white playing dice are manufactured with black dots that are indentations, as shown. So, the side with six indentations is the lightest side and the side with one indentation is the heaviest side.

You make a conjecture that when you roll a standard playing die, the number 6 will come up more often than the number 1 because 6 is the lightest side. To test your conjecture, roll a standard playing die 25 times. Record the results in the table. Does the experiment confirm your conjecture? Explain your reasoning.

EXPLORATION 2

Analyzing an Experiment
Work with a partner. To overcome the imbalance of standard playing dice, one of the authors of this book invented and patented 12-sided dice, on which each number from 1 through 6 appears twice (on opposing sides). See BigIdeasMath.com.
As part of the patent process, a standard playing die was rolled 27,090 times. The results are shown below.

What can you conclude from the results of this experiment? Explain your reasoning.

Communicate Your Answer

Question 3.
How can you use an experiment to test a conjecture?
Answer:

Question 4.
Exploration 2 shows the results of rolling a standard playing die 27,090 times to test the conjecture in Exploration 1. Why do you think the number of trials was so large?
Answer:

Question 5.
Make a conjecture about the outcomes of rolling the 12-sided die in Exploration 2. Then use the Internet to find a 12-sided die rolling simulator. Use the simulator to complete a table similar to the one shown below. How many times did you simulate rolling the die? Explain your reasoning.

Answer:

Monitoring Progress
Question 1.
Determine whether the study is a randomized comparative experiment. If it is, describe the treatment, the treatment group, and the control group. If it is not, explain why not and discuss whether the conclusions drawn from the study are valid.

Answer:
The study is a randomized comparative experiment because we have a treatment that is accident reduction, the treatment group is people who woke up before 6:30, and the control group is people who never woke up before 6:30.

Question 2.
Determine whether the following research topic is best investigated through an experiment or an observational study. Then describe the design of the experiment or observational study.
You want to know whether flowers sprayed twice per day with a mist of water stay fresh longer than flowers that are not sprayed.
Answer:
It is better to use an experimental study. The treatment will be, spraying twice per day with a mist of water, the control group will be flowers that don’t spray, and the treatment group will be flowers that will be sprayed.

Question 3.
In Example 3, the company identifies 250 people who are overweight. The subjects are randomly assigned to a treatment group or a control group. In addition, each subject is given a DVD that documents the dangers of obesity. After 3 months, most of the subjects placed in the treatment group have lost weight. Identify a potential problem with the experimental design. Then describe how you can improve it.
Answer:
The company identifies 250 people who are overweight
A potential problem is that each subject is given a DVD documenting the dangers of obesity because in that case, we have no difference between the control and treatment groups.
To improve the experiment, A DVD documenting the dangers of obesity should only be given to obese people in the treatment group.

Question 4.
You design an experiment to test the effectiveness of a vaccine against a strain of influenza. In the experiment, 100,000 people receive the vaccine and another 100,000 people receive a placebo. Identify a potential problem with the experimental design. Then describe how you can improve it.
Answer:
A potential problem is in choosing people who have been vaccinated. For example, some people may have a stronger immune system and will not need the vaccine or have a weak immune system and get sick more easily.
You can develop the experiment by randomly selecting 200,000 people with similar immune systems and forming two groups of them one of which will be vaccinated and the other will receive a placebo.

Experimental Design 11.4 Exercises

Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
Repetition of an experiment under the same or similar conditions is called _________.
Answer:
Repetition of an experiment under the same or similar conditions is called replication.

Question 2.
WRITING
Describe the difference between the control group and the treatment group in a controlled experiment.
Answer:
The treatment group is also called the experimental group. It receives treatment whose effect the researchers want to study, whereas the control group does not receive any treatment. You should include in the experiment a control group that is identical to the group that treats in every way, except that it does not receive experimental treatment.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4, determine whether the study is a randomized comparative experiment. If it is, describe the treatment, the treatment group, and the control group. If it is not, explain why not and discuss whether the conclusions drawn from the study are valid.
Question 3.

Answer:

Question 4.

Answer:
The study mentioned above is not a randomized comparative experiment since the students were not randomly assigned to the control group and treatment group.
The conclusion may or may not be valid because there may be other reasons why the can students have fewer cavities, such as better oral health practices.

ERROR ANALYSIS In Exercises 5 and 6, describe and correct the error in describing the study.

A company’s researchers want to study the effects of adding shea butter to their existing hair conditioner. They monitor the hair quality of 30 randomly selected customers using the regular conditioner and 30 randomly selected customers using the new shea butter conditioner.
Question 5.

Answer:

Question 6.

Answer:
The study cannot be an observational study because the variable is the control group because it does not receive any treatment. Thus, it is a randomized comparative experiment.

In Exercises 7–10, explain whether the research topic is best investigated through an experiment or an observational study. Then describe the design of the experiment or observational study.

Question 7.
A researcher wants to compare the body mass index of smokers and nonsmokers.
Answer:

Question 8.
A restaurant chef wants to know which pasta sauce recipe is preferred by more diners.
Answer:
A restaurant chef wants to know which pasta sauce recipe is preferred by more diners.
The treatment of different pasta sauce recipes is possible for the diner. So it is ethical to assign each dinner at the restaurant to a control or treatment group.
I will use an experimental study randomly choose one group of diners given one type of pasta sauce recipe. Randomly choose one group of diners given another type of pasta sauce recipe. Monitor each diner in both groups at dining.

Question 9.
A farmer wants to know whether a new fertilizer affects the weight of the fruit produced by strawberry plants.
Answer:

Question 10.
You want to know whether homes that are close to parks or schools have higher property values.
Answer:
Researchers continue to see that homes located near a park have an 8% to 20% increase in value compared to other properties. Thus, we cannot randomly form two groups and because of that, we will use an observational study. First, we can randomly choose homes close to the park and school and form one group and the other group will be randomly selected homes that are not close to the park and school. Find the property values for each home in both groups.

Question 11.
DRAWING CONCLUSIONS
A company wants to test whether a nutritional supplement has an adverse effect on an athlete’s heart rate while exercising. Identify a potential problem, if any, with each experimental design. Then describe how you can improve it.
a. The company randomly selects 250 athletes. Half of the athletes receive the supplement and their heart rates are monitored while they run on a treadmill. The other half of the athletes are given a placebo and their heart rates are monitored while they lift weights. The heart rates of the athletes who took the supplement significantly increased while exercising.

b. The company selects 1000 athletes. The athletes are divided into two groups based on age. Within each age group, the athletes are randomly assigned to receive the supplement or the placebo. The athletes’ heart rates are monitored while they run on a treadmill. There was no significant difference in the increases in heart rates between the two groups.
Answer:

Question 12.
DRAWING CONCLUSIONS
A researcher wants to test the effectiveness of reading novels on raising intelligence quotient (IQ) scores. Identify a potential problem, if any, with each experimental design. Then describe how you can improve it.
a. The researcher selects 500 adults and randomly divides them into two groups. One group reads novels daily and one group does not read novels. At the end of 1 year, each adult is evaluated and it is determined that neither group had an increase in IQ scores.
Answer:
A researcher selects 500 adults and randomly divides them into two groups. One group reads novels and the other doesn’t, we do not have a potential problem.

b. Fifty adults volunteer to spend time reading novels every day for 1 year. Fifty other adults volunteer to refrain from reading novels for 1 year. Each adult is evaluated and it is determined that the adults who read novels raised their IQ scores by 3 points more than the other group.

Answer:
From the given, we can note that the groups are not similar. One group is formed by the people who volunteer to read a novel and the other one doesn’t. Also, there are other habits that increase IQ scores which may have one of the volunteers. Thus the groups must be formed randomly because in that way groups will be similar and then compare the IQ scores between the groups.

Question 13.
DRAWING CONCLUSIONS
A fitness company claims that its workout program will increase vertical jump heights in 6 weeks. To test the workout program, 10 athletes are divided into two groups. The double bar graph shows the results of the experiment. Identify the potential problems with the experimental design. Then describe how you can improve it.

Answer:

Question 14.
WRITING
Explain why observational studies, rather than experiments, are usually used in astronomy.
Answer:
Astronomy is referred to as observational science and not primarily experimental science because it focuses on the acquisition of data observations of astronomical objects. The universe is still unexplored and cannot be controlled by objects or generally, in most cases, we cannot create controlled conditions, so they rather use an observational study than experiments.

Question 15.
MAKING AN ARGUMENT
Your friend wants to determine whether the number of siblings has an effect on a student’s grades. Your friend claims to be able to show causality between the number of siblings and grades. Is your friend correct? Explain.
Answer:

Question 16.
HOW DO YOU SEE IT?
To test the effect political advertisements have on voter preferences, a researcher selects 400 potential voters and randomly divides them into two groups. The circle graphs show the results of the study.

a. Is the study a randomized comparative experiment? Explain.
b. Describe the treatment.
c. Can you conclude that the political advertisements were effective? Explain.
Answer:
a. Since the voters are randomly divided into two groups, none of the groups is formed based on bias and we have a control and treatment group, it is a randomized comparative treatment.
b. The treatment was the ads for Candidate B.
c. Since there are significantly fewer differences between the graphs, we can conclude that there is no important change for candidate B.

Question 17.
WRITING
Describe the placebo effect and how it affects the results of an experiment. Explain how a researcher can minimize the placebo effect.
Answer:

Question 18.
THOUGHT PROVOKING
Make a hypothesis about something that interests you. Design an experiment that could show that your hypothesis is probably true.
Answer:
A hypothesis is essentially an idea about the population that you think might be true, but cannot prove to be true. Corn treated with the supplement is higher than corn that is not. By random selection, we form two groups with small corn plants. One group is treated with a supplement and the other is not. Keppal the other variables constant and compare the results after measure.

Question 19.
REASONING
Will replicating an experiment on many individuals produce data that are more likely to accurately represent a population than performing the experiment only once? Explain.
Answer:

Maintaining Mathematical Proficiency

Draw a dot plot that represents the data. Identify the shape of the distribution.
Question 20.
Ages: 24, 21, 22, 26, 22, 23, 25, 23, 23, 24, 20, 25
Answer:

The shape of a distribution is symmetric.

Question 21.
Golf strokes: 4, 3, 4, 3, 3, 2, 7, 5, 3, 4
Answer:

Tell whether the function represents exponential growth or exponential decay. Then graph the function.
Question 22.
y = 4^x

Answer:

x	-2	-1	0	1	2	3
y	1/16	1/4	1	4	16	64

Identify the value of the base. The base 4 is greater than 1 so the function represents exponential growth.
Plot the points on the graph from the table. Draw from left to right, a smooth curve that begins just above the x-axis, passes through the plotted points, and moves up to the right.

Question 23.
y = (0.95)^x
Answer:

Question 24.
y = (0.2)^x

Answer:
Identify the value of the base. The base 0.2 is less than 1 so the function represents exponential decay.

x	-2	-1	0	1	2	3
y	25	5	1	0.2	0.04	0.008

Draw the points from right to left, a smooth curve that begins just above the x-axis, passes through the plotted points and moves up to the left.

Question 25.
y = (1.25)^x
Answer:

Lesson 11.5 Making Inferences from Sample Surveys

Essential Question How can you use a sample survey to infer a conclusion about a population?

EXPLORATION 1

Making an Inference from a Sample
Work with a partner. You conduct a study to determine what percent of the high school students in your city would prefer an upgraded model of their current cell phone. Based on your intuition and talking with a few acquaintances, you think that 50% of high school students would prefer an upgrade. You survey 50 randomly chosen high school students and find that 20 of them prefer an upgraded model.

a. Based on your sample survey, what percent of the high school students in your city would prefer an upgraded model? Explain your reasoning.
b. In spite of your sample survey, is it still possible that 50% of the high school students in your city prefer an upgraded model? Explain your reasoning.

c. To investigate the likelihood that you could have selected a sample of 50 from a population in which 50% of the population does prefer an upgraded model, you create a binomial distribution as shown below. From the distribution, estimate the probability that exactly 20 students surveyed prefer an upgraded model. Is this event likely to occur? Explain your reasoning.

d. When making inferences from sample surveys, the sample must be random. In the situation described above, describe how you could design and conduct a survey using a random sample of 50 high school students who live in a large city.

Communicate Your Answer

Question 2.
How can you use a sample survey to infer a conclusion about a population?
Answer:
A population is a group of people or things. A sample is part of the population.
In order to make conclusions about the population, You can use the sample.
A complete set of possible cases.
A subset of the population from which data is collected.
Inference statistics is a process of using sample statistics to draw conclusions about population parameters.
In other words, the data from the sample is used to make inferences about the population.

Question 3.
In Exploration 1(c), what is the probability that exactly 25 students you survey prefer an upgraded model?
Answer:
There are 50,000 high school students
Sample = 50 high school students
The probability that exactly 25 students you survey prefer an upgraded model = 25/50 = 1/2 = 0.5
Thus the probability that exactly 25 students you survey prefer an upgraded model is 0.5

Monitoring Progress

Question 1.
The data from another random sample of 30 teen users of the social networking website are shown in the table. Estimate the population mean μ.

Answer:
The data from another random sample of 30 teen users of the social networking website are shown in the table.
To estimate the mean, we will sum all values from the table and then divide it with the count of members that participate in the sum.
μ = \(\frac { 1 }{ 30 } \)(305 + 237 + . . . + 287) = \(\frac { 1 }{ 30 } \) . 8820 = 294
So, the mean, μ = 294

Question 2.
Two candidates are running for class president. The table shows the results of four surveys of random students in the class. The students were asked whether they will vote for the incumbent. Do you think the incumbent will be reelected? Explain.

Answer:
As a sample size increases, we can notice that percentage of the percent of voters for incumbents decreases.
Thus, since for the sample size of 40, we see that support is less than 50%.
Thus the circumbent will not be rejected.

Question 3.
WHAT IF?
In Example 3, what can you conclude about the accuracy of the claim that the population proportion is 0.34 when 21 adults in your random sample say mathematics is the most valuable subject?
Answer:
21 out of 50 corresponds to a sample proportion of \(\frac { 21 }{ 50 } \) = 0.42.
In the simulation, this result occured in 5 of the 80 random samples.
It is unlikely that 21 adults out of 50 would say math is the most valuable subject when the true population percentage is 34%.
The company’s claim is probably not accurate.

Question 4.
In a survey of 1028 people in the U.S., 87% reported using the Internet. Give an interval that is likely to contain the exact percent of all people in the U.S. who use the Internet.
Answer:
Since in the survey participiate n = 1028 people from the U.S, the marigin error will be
\(\frac { 1 }{ √n } \) = \(\frac { 1 }{ √1028 } \) = 0.031
Thus, the interval that is likely to contain the exact percent of all people in the U.S who use the internet will be
(87% – 31%, 87% – 31%) = (83.9%, 90.1%)
Thus, the interval between 83.9% to 90.1%

Making Inferences from Sample Surveys 11.5 Exercises

Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
The ___________ gives a limit on how much the responses of the sample would differ from the responses of the population.
Answer: The margin of error gives a limit on how much the responses of the sample would differ from the responses of the population.

Question 2.
WRITING
What is the difference between descriptive and inferential statistics?
Answer:
Descriptive statistics uses the data to provide descriptions of the population, either through numerical calculations or graphs, or tables. Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.

Monitoring Progress and Modeling with Mathematics

Question 3.
PROBLEM SOLVING
The numbers of text messages sent each day by a random sample of 30 teen cellphone users are shown in the table. Estimate the population mean μ.

Answer:

Question 4.
PROBLEM SOLVING
The incomes for a random sample of 35 U.S. households are shown in the table. Estimate the population mean μ.

Answer:
The incomes for a random sample of 35 U.S. households are shown in the table.
The mean will be estimated as a sum of elements from the table divided with the 5. Hence
μ = \(\frac { 1 }{ 35 } \)(14300 + 72800 + . . + 63100) = $51029
So, the mean μ = 51029

Question 5.
PROBLEM SOLVING
Use the data in Exercise 3 to answer each question.
a. Estimate the population proportion ρ of teen cellphone users who send more than 70 text messages each day.
b. Estimate the population proportion ρ of teen cellphone users who send fewer than 50 text messages each day.
Answer:

Question 6.
WRITING
A survey asks a random sample of U.S. teenagers how many hours of television they watch each night. The survey reveals that the sample mean is 3 hours per night. How condent are you that the average of all U.S. teenagers is exactly 3 hours per night? Explain your reasoning.

Answer:
Not at all confident.
We have that the sample mean is 3 hours per night and we cannot expect that the average of all US teenagers is exactly 3 hours per night, maybe the result can be close to 3 but not exactly 3.

Question 7.
DRAWING CONCLUSIONS
When the President of the United States vetoes a bill, the Congress can override the veto by a two-thirds majority vote in each House. Five news organizations conduct individual random surveys of U.S. Senators. The senators are asked whether they will vote to override the veto. The results are shown in the table.

a. Based on the results of the first two surveys, do you think the Senate will vote to override the veto? Explain.
b. Based on the results in the table, do you think the Senate will vote to override the veto? Explain.
Answer:

Question 8.
DRAWING CONCLUSIONS
Your teacher lets the students decide whether to have their test on Friday or Monday. The table shows the results from four surveys of randomly selected students in your grade who are taking the same class. The students are asked whether they want to have the test on Friday.

a. Based on the results of the first two surveys, do you think the test will be on Friday? Explain.
Answer:
Based on the results of the first two surveys, the test will be on Friday because there we have more than half the majority.

b. Based on the results in the table, do you think the test will be on Friday? Explain.
Answer:
From the table, as the sample size increases, the percentage of voters for test on Friday decreases, and in the last survey, we have 45% which is not enough.

Question 9.
MODELING WITH MATHEMATICS
A national polling company claims that 54% of U.S. adults are married. You survey a random sample of 50 adults.
a. What can you conclude about the accuracy of the claim that the population proportion is 0.54 when 31 adults in your survey are married?
b. What can you conclude about the accuracy of the claim that the population proportion is 0.54 when 19 adults in your survey are married?
c. Assume that the true population proportion is 0.54. Estimate the variation among sample proportions for samples of size 50.
Answer:

Question 10.
MODELING WITH MATHEMATICS
Employee engagement is the level of commitment and involvement an employee has toward the company and its values. A national polling company claims that only 29% of U.S. employees feel engaged at work. You survey a random sample of 50 U.S. employees.
a. What can you conclude about the accuracy of the claim that the population proportion is 0.29 when 16 employees feel engaged at work?
Answer: The company’s claim is probably correct

b. What can you conclude about the accuracy of the claim that the population proportion is 0.29 when 23 employees feel engaged at work?
Answer: The company’s claim is probably incorrect
c. Assume that the true population proportion is 0.29. Estimate the variation among sample proportions for samples of size 50.
Answer: 0.15 to 0.4

In Exercises 11–16, find the margin of error for a survey that has the given sample size. Round your answer to the nearest tenth of a percent.
Question 11.
260
Answer:

Question 12.
1000
Answer:
Use the margin of error formula
The margin of error = ±\(\frac { 1 }{ n } \) = ±\(\frac { 1 }{ √1000 } \) = ±0.031
The margin of error for the survey is about ±3.1%.

Question 13.
2024
Answer:

Question 14.
6400
Answer:
Use the margin of error formula
The margin of error = ±\(\frac { 1 }{ n } \) = ±\(\frac { 1 }{√6400 } \) = ±0.0125
The margin of error for the survey is about ±1.25%.

Question 15.
3275
Answer:

Question 16.
750
Answer:
Use the margin of error formula
The margin of error = ±\(\frac { 1 }{ n } \) = ±\(\frac { 1 }{√750} \) = ±0.036
The margin of error for the survey is about ±3.6%

Question 17.
ATTENDING TO PRECISION
In a survey of 1020 U.S. adults, 41% said that their top priority for saving is retirement.
a. What is the margin of error for the survey?
b. Give an interval that is likely to contain the exact percent of all U.S. adults whose top priority for saving is retirement.
Answer:

Question 18.
ATTENDING TO PRECISION
In a survey of 1022 U.S. adults, 76% said that more emphasis should be placed on producing domestic energy from solar power.

a. What is the margin of error for the survey?
Answer:
The margin error of the sample size n = 1020 will be ±\(\frac { 1 }{ √n } \) = ±\(\frac { 1 }{ √1020 } \) = 0.031
The margin error for the survey is about ±3.1%

b. Give an interval that is likely to contain the exact percent of all U.S. adults who think more emphasis should be placed on producing domestic energy from solar power.
Answer:
To find the interval, subtract and add 3.1% to the percent of people surveyed who said that more emphasis should be placed on producing domestic energy from solar power (76%)
76% – 3.1% = 72.9%
76% + 3.1% = 79.1%
It is likely that the exact percent of all people in the U.S who said that is likely to contain the exact percent of all U.S adults whose top priority for saving is retirement is between 72.9% and 79.1%

Question 19.
ERROR ANALYSIS
In a survey, 8% of adult Internet users said they participate in sports fantasy leagues online. The margin of error is ±4%. Describe and correct the error in calculating the sample size.

Answer:

Question 20.
ERROR ANALYSIS
In a random sample of 2500 consumers, 61% prefer Game A over Game B. Describe and correct the error in giving an interval that is likely to contain the exact percent of all consumers who prefer Game A over Game B.

Answer:
In a random sample of 2500 consumers, 61% prefer Game A over Game B.
To find, the interval subtracts and add margin error to the percent of people surveyed who prefer Game A over Game B (61%). Thus since the margin error is 2%, the interval is 61% – 2% = 59%
61% + 2% = 63%
hence, it is likely that the exact percent of all consumers who prefer Game A over Game B is between 59% and 63%.

Question 21.
MAKING AN ARGUMENT
Your friend states that it is possible to have a margin of error between 0 and 100 percent, not including 0 or 100 percent. Is your friend correct? Explain your reasoning.
Answer:

Question 22.
HOW DO YOU SEE IT?
The figure shows the distribution of the sample proportions from three simulations using different sample sizes. Which simulation has the least margin of error? the greatest? Explain your reasoning.

Answer:
The margin error is proportional to the standard deviation. Thus, from the graph, we can conclude that simulation a has the least standard deviation which implies a has the least marginal error.
The greatest standard deviation has simulation c which implies that simulation c has the greatest marginal error.
The least margin of error has simulation a. The greatest margin of error has simulation c.

Question 23.
REASONING
A developer claims that the percent of city residents who favor building a new football stadium is likely between 52.3% and 61.7%. How many residents were surveyed?

Answer:

Question 24.
ABSTRACT REASONING
Suppose a random sample of size n is required to produce a margin of error of ±E. Write an expression in terms of n for the sample size needed to reduce the margin of error to ±1—2E. How many times must the sample size be increased to cut the margin of error in half? Explain.
Answer:
E = ±\(\frac { 1 }{ √n } \)
To compute how many times must the sample size be increased to cut the margin of error in half
\(\frac { 1 }{ 2 } \)E = ±\(\frac { 1 }{ √x } \)
\(\frac { 1 }{ 2 } \) . (±\(\frac { 1 }{ √n } \)) = ±\(\frac { 1 }{ √x } \)
(±\(\frac { 1 }{ 2√n } \))² = (±\(\frac { 1 }{ √x } \))²
\(\frac { 1 }{ 4n } \) = \(\frac { 1 }{ x } \)
x = 4n
The sample must be four times greater than the size of the sample n with the margin error E.

Question 25.
PROBLEM SOLVING
A survey reported that 47% of the voters surveyed, or about 235 voters, said they voted for Candidate A and the remainder said they voted for Candidate B.
a. How many voters were surveyed?
b. What is the margin of error for the survey?
c. For each candidate, find an interval that is likely to contain the exact percent of all voters who voted for the candidate.
d. Based on your intervals in part (c), can you be confident that Candidate B won? If not, how many people in the sample would need to vote for Candidate B for you to be condent that Candidate B won? (Hint: Find the least number of voters for Candidate B so that the intervals do not overlap.)
Answer:

Question 26.
THOUGHT PROVOKING
Consider a large population in which ρ percent (in decimal form) have a certain characteristic. To be reasonably sure that you are choosing a sample that is representative of a population, you should choose a random sample of n people where n > 9\(\left(\frac{1-\rho}{\rho}\right)\) .
a. Suppose ρ= 0.5. How large does n need to be?
Answer:
If the proportion population is p = 0.5, let’s compute the size of the population
n > 9(\(\frac { 1 – p }{ p } \)
For p = 0.5
n > 9(\(\frac { 1 – 0.5 }{ 0.5 } \)
n > 9(0.5/0.5)
n > 9

b. Suppose ρ= 0.01. How large does n need to be?
Answer:
If proportion population p = 0.01
n > 9(\(\frac { 1 – 0.01 }{ 0.01 } \)
n > 9 . (\(\frac { 0.99 }{ 0.01 } \))
n > 9 . 99
n > 891

c. What can you conclude from parts (a) and (b)?
Answer:
From parts a and b, you can conclude that the smaller percentage of people with a certain characteristic, the larger the sample size is necessary for the appropriate representation of the population.

Question 27.
CRITICAL THINKING
In a survey, 52% of the respondents said they prefer sports drink X and 48% said they prefer sports drink Y. How many people would have to be surveyed for you to be confident that sports drink X is truly preferred by more than half the population? Explain.
Answer:

Maintaining Mathematical Proficiency

Find the inverse of the function.
Question 28.
y= 10^x-3

Answer:
Given,
y= 10^x-3
To find the inverse function, interchange x and y
x = y
x = 10^y-3
logx = log10^y-3
logx = y – 3
y = logx + 3

Question 29.
y= 2^x – 5
Answer:

Question 30.
y= ln (x+ 5)

Answer:
Given,
y= ln (x+ 5)
To find the inverse function, interchange x and y
x = y
x = ln(y + 5)
e^x = e^ln(y+5)
e^x = y + 5
y = e^x – 5

Question 31.
y= log₆x− 1
Answer:

Determine whether the graph represents an arithmetic sequence or a geometric sequence. Then write a rule for the nth term.
Question 32.

Answer:
From the graph, we have the following points (1, 17), (2, 14), (3, 11), (4, 8)
Now, let’s denote the following participants of the sequence
a1 = 17
a2 = 14
a3 = 11
a4 = 8
Now, to decide whether to do is an arithmetic or geometric sequence, let’s consider the following terms
d = a<n-1> – a<n-2> = a<k-1> – a<k-2>
q = \(\frac { a<n> }{ a<n-1> } \) = \(\frac { a<k> }{ a<k-1> } \)
a2 – a1 = a3 – a2 = a3 – a4 = -3
a<n> = a<1> + (n – 1)d
Hence, by using the formula, the form of the sequence is a<n> = 17 + (n – 1)(-3)
= 17 – 3n + 3
= 20 – 3n

Question 33.

Answer:

Question 34.

Answer:
From the graph, we have the following points (1, 32), (2, 16), (3, 8), (4, 4)
Now, let’s denote the following participants of the sequence
a1 = 32
a2 = 16
a3 = 8
a4 = 4
Now, to decide whether to do is an arithmetic or geometric sequence, let’s consider the following terms
d = a<n-1> – a<n-2> = a<k-1> – a<k-2>
r = \(\frac { an }{ a(n – 1) } \)
\(\frac { a2 }{ a1 } \) = \(\frac { 1 }{ 2 } \)
a<n> = a1 . r^(n-1)
Hence, by using the formula, the form of the sequence is a<n> = 24. (\(\frac { 1 }{ 2 } \))^(n-1)
= 24 . \(\frac { 1 }{ 2^(n-1) } \)

Lesson 11.6 Making Inferences from Experiments

Essential Question How can you test a hypothesis about an experiment?

EXPLORATION 1

Resampling Data
Work with a partner. A randomized comparative experiment tests whether water with dissolved calcium affects the yields of yellow squash plants. The table shows the results.

a. Find the mean yield of the control group and the mean yield of the treatment group. Then find the difference of the two means. Record the results.
b. Write each yield measurement from the table on an equal-sized piece of paper. Place the pieces of paper in a bag, shake, and randomly choose 10 pieces of paper. Call this the “control” group, and call the 10 pieces in the bag the “treatment” group. Then repeat part (a) and return the pieces to the bag. Perform this resampling experiment five times.
c. How does the difference in the means of the control and treatment groups compare with the differences resulting from chance?

EXPLORATION 2

Evaluating Results
Work as a class. To conclude that the treatment is responsible for the difference in yield, you need strong evidence to reject the hypothesis:
Water dissolved in calcium has no effect on the yields of yellow squash plants.
To evaluate this hypothesis, compare the experimental difference of means with the resampling differences.
a. Collect all the resampling differences of means found in Exploration 1(b) for the whole class and display these values in a histogram.
b. Draw a vertical line on your class histogram to represent the experimental difference of means found in Exploration 1(a).
c. Where on the histogram should the experimental difference of means lie to give evidence for rejecting the hypothesis?
d. Is your class able to reject the hypothesis? Explain your reasoning.

Communicate Your Answer

Question 3.
How can you test a hypothesis about an experiment?
Answer:
1. Select your research area as a null hypothesis and an alternate hypothesis.
2. Collect the data designed to test the hypothesis.
3. Next perform an appropriate statistical test.
4. Describe whether to reject your null hypothesis.
5. Write the findings in your results and discussion section.

Question 4.
The randomized comparative experiment described in Exploration 1 is replicated and the results are shown in the table. Repeat Explorations 1 and 2 using this data set. Explain any differences in your answers.

Answer:

Monitoring Progress

Question 1.
In Example 1, interpret the meaning of \(\bar{x}_{\text {treatment }}\) \(\bar{x}_{\text {control }}\) control when the difference is (a) negative, (b) zero, and (c) positive.
Answer:
a. Positive means the treatment increases the total yield of cherry tomato plants.
b. Negative means the treatment decreases the total yield of cherry tomato plants.
c. Zero means the treatment made no difference.

Question 2.
In Example 3, what are the consequences of concluding that the hypothesis is false when it is actually true?
Answer:
a. The mean score of the control group = \(\frac { 49 + 45 + 43 + 47 + 46 + 45 + 47 + 46 }{ 8 } \)
= \(\frac { 368 }{ 8 } \)
= 46
b. The mean score of the treatment group = \(\frac { 39 + 40 + 39 + 37 + 41 + 40 + 42 + 43 }{ 8 } \)
= \(\frac { 281 }{ 8 } \)
= 35.13
c. The experimental difference of the means = 35.13 – 46
= -10.87
d. The double data plot of the data is

e. The data shows that the treatment groups show the lowest scores compared to the control group. We can conclude that music therapy may be effective for college students with depression.

Making Inferences from Experiments 11.6 Exercises

Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
A method in which new samples are repeatedly drawn from the data set is called ____________.
Answer:
A method in which new samples are repeatedly drawn from the data set is called resampling.

Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

Answer:
The experimental difference of the means between the averages of the experimental (treatment) group and control groups, which can be written as μ<treatment> – μ<control>
So, the different question is the third question which asks to determine the square root of the average of the squared differences from 2.85.
μ<treatment> = \(\frac { 1 }{ 6 } \)(0.4 + 0.6 + 0.5 + 0.6 + 0.7+ 0.5)
= 0.55
μ<control> = \(\frac { 1 }{ 6 } \)(3.3 + 3.2 + 3.7 + 3.5 + 3.3 + 3.4) = 3.4
μ<treatment> – μ<control> = 0.55 – 3.4 = -2.85
The same answer is -2.85
The answer to the different questions is
= √(\(\frac { 1 }{ 12 } \)((3.3 – (-2.85))² + (3.2 – (-2.85))² + . . . +(0.5 – (-2.85))²))
= √(\(\frac { 1 }{ 12 } \)(303.95))
= √(25.32)
= 5.03
The different answer is 5.03

Monitoring Progress and Modeling with Mathematics

Question 3.
PROBLEM SOLVING
A randomized comparative experiment tests whether music therapy affects the depression scores of college students. The depression scores range from 20 to 80, with scores greater than 50 being associated with depression. The control group has eight students and the treatment group, which receives the music therapy, has eight students. The table shows the results.

a. Find the mean score of the control group.
b. Find the mean score of the treatment group.
c. Find the experimental difference of the means.
d. Display the data in a double dot plot.
e. What can you conclude?

Answer:

Question 4.
PROBLEM SOLVING
A randomized comparative experiment tests whether low-level laser therapy affects the waist circumference of adults. The control group has eight adults and the treatment group, which receives the low-level laser therapy, has eight adults. The table shows the results.

a. Find the mean circumference of the control group.
Answer:
By using the given data, let’s find the mean circumference of the control group.
μ<control> = \(\frac { 1 }{ 8 } \)(34.6 + 35.4 + 33 + 34.6 + 35.2 + 35.2 + 36.2 + 35) = \(\frac { 1 }{ 8 } \)(279.2) = 34.9

b. Find the mean circumference of the treatment group.
Answer:
The mean of the treatment group
μ<treatment> = \(\frac { 1 }{ 8 } \)(31.4 + 33 + 32.4 + 32.6 + 33.4 + 33.4 + 33) = \(\frac { 1 }{ 8 } \)(264) = 33

c. Find the experimental difference of the means.
Answer:
The experimental mean is
μ<treatment> – μ<control> = 33 – 34.9 = -1.9

d. Display the data in a double dot plot.
Answer:
The dot plot of the control and treatment groups

e. What can you conclude?
Answer:
We can conclude that low-level laser therapy has little effect in reducing waist circumference in adults.

Question 5.
ERROR ANALYSIS
In a randomized comparative experiment, the mean score of the treatment group is 11 and the mean score of the control group is 16. Describe and correct the error in interpreting the experimental difference of the means.

Answer:

Question 6.
REASONING
In Exercise 4, interpret the meaning of \(\bar{x}_{\text {treatment }}-\bar{x}_{\text {control }}\) when the difference is positive, negative, and zero.
Answer:
The mean waist circumference of the treatment group > the mean waist circumference of the control group.
Positive means the treatment made the waist circumference larger.
The mean waist circumference of the treatment group < the mean waist circumference of the control group.
Negative means the treatment made the waist circumference smaller.
The mean waist circumference of the treatment group is the same as the mean waist circumference of the control group.
Zero means the treatment made no difference.

Question 7.
MODELING WITH MATHEMATICS
Resample the data in Exercise 3 using a simulation. Use the means of the new control and treatment groups to calculate the difference of the means.
Answer:

Question 8.
MODELING WITH MATHEMATICS
Resample the data in Exercise 4 using a simulation. Use the means of the new control and treatment groups to calculate the difference of the means.
Answer:
First, combine the measurements from both groups and assign a number to each value. Let the numbers 1 through 8 represent the data in the original control group, and let the numbers 9 through 16 represent the data in the original treatment group

Original control group	34.6	35.4	33	34.6	35.2	35.2	36.2	35
assigned number	1	2	3	4	5	6	7	8

Original treatment group	31.4	33	32.4	32.6	33.4	33.4	33.8	33
assigned number	9	10	11	12	13	14	15	16

Use a random generator. Randomly generate 16 numbers from 1 through 16 without repeating a number.

8	5	15	7	12	1	13	3
14	4	16	9	11	10	2	6

Use the first 8 numbers to make the new control group, and the next 8 to make the new treatment group.

New control group	35	35.2	34.8	36.2	32.6	34.6	33.4	33
New treatment group	33.4	34.6	33	31.4	32.4	33	35.4	35.2

The mean of the new control and new treatment groups are
μ<new control> = \(\frac { 1 }{ 8 } \)(35 + 35.2 + 34.8 + 36.2 + 32.6 + 34.6 + 33.4 + 33)
= 34.4
μ<new treatment> = \(\frac { 1 }{ 8 } \)(33.4 + 34.6 + 33 + 31.4 + 32.4 + 33 + 35.4 + 35.2)
= 33.6
The experimental difference = μ<new treatment> – μ<new control> = 33.6 – 34.4 = -0.8
This is greater than the experimental difference found in example 4.

Question 9.
DRAWING CONCLUSIONS
To analyze the hypothesis below, use the histogram which shows the results from 200 resamplings of the data in Exercise 3.
Music therapy has no effect on the depression score.
Compare the experimental difference in Exercise 3 with the resampling differences. What can you conclude about the hypothesis? Does music therapy have an effect on the depression score?

Answer:

Question 10.
DRAWING CONCLUSIONS
Suppose the experimental difference of the means in Exercise 3 had been −0.75. Compare this experimental difference of means with the resampling differences in the histogram in Exercise 9. What can you conclude about the hypothesis? Does music therapy have an effect on the depression score?
Answer:
The histogram is approximately bell-shaped and fairly symmetric, so the differences have an approximately normal distribution. Note that the hypothesis assumes that the difference in the mean yields is 0.
The experimental difference of -0.75, however, lies close to the left tail.
From the graph, there are about 106 to 113 values out of 200 that are greater than -0.75 which is more than 50% of the values.
Also, the experimental difference falls inside the middle 90% of the resampling differences. It is likely to get a difference this large when you assume that the difference is 0, suggesting the control group and the treatment group is not different.
You can conclude that the hypothesis is most likely true. Thus, music therapy has no effect on depression scores.

Question 11.
WRITING
Compare the histogram in Exercise 9 to the histogram below. Determine which one provides stronger evidence against the hypothesis, Music therapy has no effect on the depression score. Explain.

Answer:

Question 12.
HOW DO YOU SEE IT?
Without calculating, determine whether the experimental difference, \(\bar{x}_{\text {treatment }}-\bar{x}_{\text {control }}\), is positive, negative, or zero. What can you conclude about the effect of the treatment? Explain.

Answer:
Most of the control group values are greater than the treatment group values which implies that the mean of the control group values will be greater than the mean of the treatment group values.
The experimental difference is μ<treatment> – μ<control>
The experimental difference is negative. Also, by using the given graph, since the difference between the control values is treatment values is little.
We have a slight effect.

Question 13.
MAKING AN ARGUMENT
Your friend states that the mean of the resampling differences of the means should be close to 0 as the number of resamplings increase. Is your friend correct? Explain your reasoning.
Answer:

Question 14.
THOUGHT PROVOKING
Describe an example of an observation that can be made from an experiment. Then give four possible inferences that could be made from the observation.
Answer:
The 4 possible inferences are as follows,
1. The hair growth shampoo has no effect on hair growth.
2. The hair growth shampoo causes decreases in hair growth.
3. The hair growth shampoo causes a slight increase in the hair growth
4. The hair growth shampoo causes a significant increase in hair growth.

Question 15.
CRITICAL THINKING
In Exercise 4, how many resamplings of the treatment and control groups are theoretically possible? Explain.
Answer:

Maintaining Mathematical Proficiency

Factor the polynomial completely.
Question 16.
5x³ − 15x²

Answer:
Given,
5x³ − 15x²
Take 5x² as common.
5x³ − 15x²  = 5. x² . x – 5. x² . 3
= 5x²(x – 3)
Thus, 5x³ − 15x² = 5x²(x – 3)

Question 17.
y³ − 8
Answer:

Question 18.
z³ + 5z² − 9z − 45
Answer:
Given,
z³ + 5z² − 9z − 45
z³ + 5z² − 9z − 45 = z³ + 5z² – 9z – 45 + 6z – 6z
= z³ + 5z² + 6z – 15z – 45
= z(z² + 5z + 6) – 15(z + 3)
z² + 5z + 6 = 0
Using the quadratic formula
z₁,₂ = \(\frac { -5 ± √(25 – 4 . 6)}{ 2 . 1 } \)
= \(\frac { -5 ± √(25 – 24)}{ 2} \)
= \(\frac { -5 ± 1}{ 2 } \)
The solutions are \(\frac { -5 + 1 }{ 2 } \) = -2, \(\frac { -5 – 1 }{ 2 } \) = -3
The fcators are
z² + 5z + 6 = (z + 2)(z + 3)
z³ + 5z² − 9z − 45 = z(z² + 5z + 6) – 15(z + 3)
= z(z + 2)(z + 3) – 15(z + 3)
= (z +3)(z² + 2z – 15)
z² + 2z – 15 = 0
Using the quadratic formula
z₁,₂ = \(\frac { -2 ± √(4 – 4 . (-15))}{ 2 . 1 } \)
= \(\frac { -2 ± √(4 + 60)}{ 2 } \)
= \(\frac { -2 ± 8}{ 2} \)
Thus, the solutions are \(\frac { -2 + 8}{ 2} \) = 3, \(\frac { -2 – 8}{ 2} \) = -5
z² + 2z – 15 = (z – 3)(z + 5)
(z +3)(z² + 2z – 15) = (z+3)(z – 3)(z + 5)

Question 19.
81w⁴ − 16
Answer:

Determine whether the inverse of f is a function. Then find the inverse.
Question 20.
f(x) = \(\frac{3}{x+5}\)
Answer:

From the graph
y = \(\frac { 3 }{ x + 5 } \)
x = \(\frac { 3 }{ y + 5 } \)
x(y + 5) = 3
y + 5 = \(\frac { 3 }{ x } \)
y = \(\frac { 3 }{ x } \) – 5

Question 21.
f(x) = \(\frac{1}{2x-1}\)
Answer:

Question 22.
f(x) = \(\frac{2}{x}\) − 4
Answer:

From the graph
y = \(\frac { 2 }{ x } \) – 4
x = \(\frac { 2 }{ y } \) – 4
x + 4 = \(\frac { 2 }{ y } \)
y(x + 4) = 2
y = \(\frac { 2 }{ x + 4 } \)

Question 23.
f(x) = \(\frac{3}{x^{2}}\) + 1
Answer:

Data Analysis and Statistics Performance Task: Curving the Test

11.4–11.6 What Did You Learn?

Core Vocabulary

Core Concepts
Section 11.4
Randomization in Experiments and Observational Studies, p. 621
Comparative Studies and Causality, p. 621
Analyzing Experimental Designs, p. 622

Section 11.5
Estimating Population Parameters, p. 626
Analyzing Estimated Population Parameters, p. 628

Section 11.6
Experiments with Two Samples, p. 634
Resampling Data Using Simulations, p. 635
Making Inferences About Treatments, p. 636

Mathematical Practices
Question 1.
In Exercise 7 on page 623, find a partner and discuss your answers. What questions should you ask your partner to determine whether an observational study or an experiment is more appropriate?
Answer:

Question 2.
In Exercise 23 on page 632, how did you use the given interval to find the sample size?
Answer:

Performance Task: Curving the Test
Test scores are sometimes curved for different reasons using different techniques. Curving began with the assumption that a good test would result in scores that were normally distributed about a C average. Is this assumption valid? Are test scores in your class normally distributed? If not, how are they distributed? Which curving algorithms preserve the distribution and which algorithms change it?
To explore the answers to these questions and more, go to BigIdeasMath.com.

Data Analysis and Statistics Chapter Review

11.1 Using Normal Distributions (pp. 595–602)

Question 1.
A normal distribution has mean μ and standard deviation σ. An x-value is randomly selected from the distribution. Find P(x ≤ μ − 3σ).
Answer:
Find the probability P(x ≤ μ – 3σ)
P(x ≤ μ – 3σ) = \(\frac { 1 – 0.997 }{ 2 } \) = \(\frac { 0.003 }{ 2 } \)
= 0.0015
The probability is 0.0015

Question 2.
The scores received by juniors on the math portion of the PSAT are normally distributed with a mean of 48.6 and a standard deviation of 11.4. What is the probability that a randomly selected score is at least 76?
Answer:
The scores received by juniors on the math portion of the PSAT are normally distributed with μ = 48.6, σ = 11.4
z = \(\frac { x – μ }{ σ } \)
P(x ≥76) = P(\(\frac { x – 48.6 }{ 11.4 } \)≥\(\frac { 76 – 48.6 }{ 11.4 } \))
= P(z≥2.4) = 1 – P(z≤2.4)
= 1 – 0.9918
= 0.0082

11.2 Populations, Samples, and Hypotheses (pp. 603−608)

Question 3.
To estimate the average number of miles driven by U.S. motorists each year, a researcher conducts a survey of 1000 drivers, records the number of miles they drive in a year, and then determines the average. Identify the population and the sample.
Answer:
As the survey does not participate all motorists from the US, we can conclude that in the population of all motorist drivers in the US, a sample of 1000 drivers is surveyed.

Question 4.
A pitcher throws 40 fastballs in a game. A baseball analyst records the speeds of 10 fastballs and finds that the mean speed is 92.4 miles per hour. Is the mean speed a parameter or a statistic? Explain.
Answer:
A pitcher throws 40 fastballs in a game. A baseball analyst records the speeds of 10 fastballs and finds that the mean speed is 92.4 miles per hour
Since we have an average speed of 10 fastballs of some pitchers, the mean value is calculated from a sample, not from the entire population. It represents statistics. Thus, this average represents statistics.

Question 5.
A prize on a game show is placed behind either Door A or Door B. You suspect the prize is more often behind Door A. The show host claims the prize is randomly placed behind either door. What should you conclude when the prize is behind Door A for 32 out of 50 contestants?
Answer:
A prize on a game show is placed behind either Door A or Door B.
Since more than half of the prizes for the customers are behind door A, the host’s claim that the prize is randomly placed behind either door is most likely false.

11.3 Collecting Data (pp. 609−616)

Question 6.
A researcher wants to determine how many people in a city support the construction of a new road connecting the high school to the north side of the city. Fifty residents from each side of the city are surveyed. Identify the type of sample described and determine whether the sample is biased.
Answer:
Fifty residents from each side of the city are surveyed to determine how many people in a city support the construction of a new road connecting the high school to the north side of the city. Thus, it is a stratified sample.

Question 7.
A researcher records the number of people who use a coupon when they dine at a certain restaurant. Identify the method of data collection.
Answer:
It is an observational study because the researcher simply observes the subjects without interfering.
The method of data collection is an observational study.

Question 8.
Explain why the survey question below may be biased or otherwise introduce bias into the survey. Then describe a way to correct the flaw.
“Do you think the city should replace the outdated police cars it is using?”
Answer:
The answer is biased because it is set in a way that encourages the answer yes. The question can be set as “Do you think the police should replace cars?”

11.4 Experimental Design (pp. 619–624)

Question 9.
A restaurant manager wants to know which type of sandwich bread attracts the most repeat customers. Is the topic best investigated through an experiment or an observational study? Describe how you would design the experiment or observational study.
Answer:
We can do an experimental study on this problem. For example, suppose we have 5 types of sandwich bread available.
Then we can make 5 randomly selected groups with the same number of people and sell each group a sandwich with a certain type of bread.
For each group, note how many people came again to buy the same sandwich.

Question 10.
A researcher wants to test the effectiveness of a sleeping pill. Identify a potential problem, if any, with the experimental design below. Then describe how you can improve it.
The researcher asks for 16 volunteers who have insomnia. Eight volunteers are given the sleeping pill and the other 8 volunteers are given a placebo. Results are recorded for 1 month.
Answer:
As the participants in the study are not randomly selected, they cannot represent the whole population. Thus, we must randomly select the members of the population for the study.

Question 11.
Determine whether the study is a randomized comparative experiment. If it is, describe the treatment, the treatment group, and the control group. If it is not, explain why not and discuss whether the conclusions drawn from the study are valid.

Answer:
It is a randomized comparative experiment because customers are randomly assigned to the control group or the treatment group. The treatment is a new design of a car wash.
The treatment group is the group that uses the new design to wash their cars and the control group will be a group that uses the old design to wash their cars.

11.5 Making Inferences from Sample Surveys (pp. 625−632)

Question 12.
In a survey of 1017 U.S. adults, 62% said that they prefer saving money over spending it. Give an interval that is likely to contain the exact percent of all U.S. adults who prefer saving money over spending it.
Answer:
Given,
In a survey of 1017 U.S. adults, 62% said that they prefer saving money over spending it.
n = 1017 U.S adults,
The margin error will be
±\(\frac { 1 }{ √n } \) = ±\(\frac { 1 }{ √1017 } \) = ±0.031
Thus, the interval that is likely to contain the exact percentage of all adults from the U.S who prefer saving money over spending it will be
(62% + 3.1%, 62% – 3.1%) = (65.1%, 58.9%)
Thus, the interval is between 65.1% t0 58.9%

Question 13.
There are two candidates for homecoming king. The table shows the results from four random surveys of the students in the school. The students were asked whether they will vote for Candidate A.Do you think Candidate A will be the homecoming king? Explain.

Answer:
When the sample size increases, it approaches 46.8%. Since that is less than 50% it is not as likely as the probability of candidate B winning homecoming king.
A sample size of 62 has 46.8% of voters saying yes. As the sample size increases, the percentage decreases.

11.6 Making Inferences from Experiments (pp. 633−638)

Question 14.
Describe how to use a simulation to resample the data in the example above. Explain how this allows you to make inferences about the data when the sample size is small.
Answer:
When the sample size is small, we can repeat this process more times and make new control and new treatment groups, and see how often you get differences between the new groups that are at least as large as the one you measured.
Combine the measurements from both groups and assign a number to each value. Let the numbers 1 through 10 represent the data in the original control group
Let the numbers 11 through 20 represent the data in the original treatment group

Control group	4.5	4.5	4.8	4.4	4.4	4.7	4.3	4.5	4.1	4.2
Assigned number	1	2	3	4	5	6	7	8	9	10

Treatment group	4.6	4.8	5.0	4.8	4.7	4.6	4.9	4.9	4.8	4.4
Assigned number	11	12	13	14	15	16	17	18	19	20

Use a random generator. Randomly generate 20 numbers from 1 through 20 without repeating a number.

5	1	15	13	18	17	4	20	10	8
9	2	12	16	6	19	11	3	7	14

Use the first 10 numbers to make the new control group, and the next 10 to make the new treatment group.

New control group	4.4	4.5	4.7	5.0	4.9	4.9	4.4	4.4	4.2	4.5
New treatment group	4.1	4.5	4.8	4.6	4.7	4.8	4.6	4.8	4.3	4.8

Now let’s find the mean yields of the new control and treatment groups
μ<new control> = \(\frac { 1 }{ 10 } \)(4.4 + 4.5 + 4.7 + 5.0 + 4.9 + 4.9 + 4.4 + 4.4 + 4.2 + 4.5)
= 4.59
μ<new treatment> = \(\frac { 1 }{ 10 } \)(4.1 + 4.5 + 4.8 + 4.6 + 4.7 + 4.8 + 4.6 + 4.8 + 4.3 + 4.8)
= 4.6
So, μ<new treatment> – μ<new control> = 4.6 – 4.59 = 0.01

Data Analysis and Statistics Chapter Test

Question 1.
Market researchers want to know whether more men or women buy their product. Explain whether this research topic is best investigated through an experiment or an observational study. Then describe the design of the experiment or observational study.
Answer:
For the researcher, it is best investigated through an observational study because random samples of customers can be selected for the groups being studied. For example, we can record every th customer who buys the product and record if he was women or men.

Question 2.
You want to survey 100 of the 2774 four-year colleges in the United States about their tuition cost. Describe a method for selecting a random sample of colleges to survey.
Answer:
Of the four-year colleges in the United States, we must randomly select colleges. Therefore, we assign one integer to each college. Now, using a random integer generator, we can select numbers from to and make a random sample of 100 colleges that match a random sample of numbers.

Question 3.
The grade point averages of all the students in a high school are normally distributed with a mean of 2.95 and a standard deviation of 0.72. Are these numerical values parameters or statistics? Explain.
Answer:
Since these values represent the mean and standard deviation of all the students in a high school, these values represent paramenters because they are calculated from the entire population.

A normal distribution has a mean of 72 and a standard deviation of 5. Find the probability that a randomly selected x-value from the distribution is in the given interval.
Question 4.
between 67 and 77
Answer:
The normal distribution has a mean of 72 and a standard deviation of 5.
z = \(\frac { x – μ }{ σ } \)
P(67≤x≤77) = P(\(\frac { 67 – 72 }{ 5 } \)≤ \(\frac { x – 72 }{ 5 } \) ≤\(\frac { 77 – 72 }{ 5 } \))
= P(-1≤z≤1) = P(z<1) – P(z<-1)
= 0.8413 – 0.1587 = 0.6826
The probability is 0.68

Question 5.
at least 75
Answer:
The normal distribution has a mean of 72 and a standard deviation of 5.
82 – μ = 82 – 72 = 10 = 2 . 5 = 2σ
10 = μ – 2σ
P(x≤82) = P(x≤μ – 2σ) = 0.5 + \(\frac { 0.95 }{ 2 } \)
= 0.5 + 0.475 = 0.975
The probability is 0.975

Question 6.
at most 82
Answer:
The normal distribution has a mean of 72 and a standard deviation of 5.
z = \(\frac { x – μ }{ σ } \)
P(x≥75) = P(\(\frac { x – 72 }{ 5 } \)≥ \(\frac { 75 – 72 }{ 5 } \))
= P(z≥0.6) = 1 – P(z< 0.6)
= 1 – 0.7257
= 0.2743
The probability is 0.2743

Question 7.
A researcher wants to test the effectiveness of a new medication designed to lower blood pressure. Identify a potential problem, if any, with the experimental design. Then describe how you can improve it.
The researcher identifies 30 people with high blood pressure. Fifteen people with the highest blood pressure are given the medication and the other 15 are given a placebo. After 1 month, the subjects are evaluated.

Answer:
Since 15 people with the highest blood pressure are assigned to one group, we can notice that people are not randomly assigned. Thus, to fix this problem, we can randomly form two groups of 15 people, and then one group will take the medicine and another one will take a placebo.

Question 8.
A randomized comparative experiment tests whether a vitamin supplement increases human bone density (in grams per square centimeter). The control group has eight people and the treatment group, which receives the vitamin supplement, has eight people. The table shows the results.

a. Find the mean yields of the control group, \(\bar{x}_{\text {control }}\), and the treatment group, \(\bar{x}_{\text {treatment }}\).
b. Find the experimental difference of the means, \(\bar{x}_{\text {treatment }}-\bar{x}_{\text {control }}\).
c. Display the data in a double dot plot. What can you conclude?
d. Five hundred resamplings of the data are simulated. Out of the 500 resampling differences, 231 are greater than the experimental difference in part (b). What can you conclude about the hypothesis, The vitamin supplement has no effect on human bone density? Explain your reasoning.
Answer:
a. μ<control> = \(\frac { 1 }{ 8 } \)(0.9 + 1.2 + 1.0 + 0.8 + 1.3 + 1.1 + 0.9 + 1.0)
= 1.025
μ<treatment> = \(\frac { 1 }{ 8 } \)(1.2 + 1.0 + 0.9 + 1.3 + 1.2 + 0.9 + 1.3 + 1.2)
= 1.125
b. μ<treatment> – μ<control> = 1.125 – 1.025 = 0.1
c. The double dot plot is

d. The hypothesis which claims that vitamin supplement has no effect on human bone density assumes that the difference of the means is equal to 0. Now, from 500 resampling differences, 231 are greater than the experimental differences in part b, which is 0.1. Now this implies that more than half of the resamplings are less than 0.1, which is less than one SD. So, the hypothesis is probably true.

Question 9.
In a recent survey of 1600 randomly selected U.S. adults, 81% said they have purchased a product online.

a. Identify the population and the sample. Describe the sample.
Answer:
In the population of all U.S adults a sample of 1600 randomly selected adults have participated in the survey

b. Find the margin of error for the survey.
Answer:
Since in the survey participate n = 1600 U.S adults, the margin error will be
±\(\frac { 1 }{ √n } \) = ±\(\frac { 1 }{ √1600 } \) = ±0.025
The margin error will be ±2.5%

c. Give an interval that is likely to contain the exact percent of all U.S. adults who have purchased a product online.
Answer:
Thus the interval that is likely to contain the exact percentage of all adults from the U.S who said that they have purchased a product online will be
(81% – 2.5%, 81% + 2.5%) = (78.5%, 83.5%)
The interval is between 78.5% to 83.5%.

d. You survey 75 teachers at your school. The results are shown in the graph. Would you use the recent survey or your survey to estimate the percent of U.S. adults who have purchased a product online? Explain.
Answer:
You survey 75 teachers at your school.
Since we surveyed only the teachers from one school we cannot use this survey because it cannot represent the entire population. So, we will use a recent survey.

Data Analysis and Statistics Cumulative Assessment

Question 1.
Your friend claims any system formed by three of the following equations will have exactly one solution.

a. Write a linear system that would support your friend’s claim.
b. Write a linear system that shows your friend’s claim is incorrect.
Answer:
a. Each of the given equations represents a plane equation. A system of 3 equations will have one solution if these planes intersect at one point.
3x + y + 3z = 6 — (i)
x + y + z = 2 — (ii)
3x + y + 9z = 12 — (iii)
We can write 2nd equation as y = 2 – x – z
Solve 1 & 3,
3x + 2 – x – z + 3z = 6
2x + 2z = 4 — (iv)
3x + 2 – x – z +9z = 12
2x + 8z = 10 — (v)
Subtract (iv) & (v)
2x + 8z – 2x – 2z = 10 – 4
6z = 6
z = 1
Put z = 1 in (iv)
2x + 2 = 4
x = 1
Put x, z values in y = 2 – x – z
y = 2 – 1 – 1
y = 0
Hence, the solution of the system is (x, y, z) = (1, 0, 1) and the friend’s calim is correct. There is only one solution to the system.
b. To prove thae claim is incorrect, take the following equations
3x + y + 3z = 6 — (i)
x + y + z = 2 — (ii)
4x – 2y + 4z = 8 — (iii)
We can write 2nd equation as y = 2 – x – z
Change the first & third equations
3x + 2 – x – z + 3z = 6
2x + 2z = 4
x + z = 2— (iv)
4x – 2(2 – x – z) + 4z = 8
4x – 4 + 2x + 2z + 4z = 8
6x + 6z = 12
x + z = 2 — (v)
As both (iv), v are same, the equation has many solutions
y = 2 – 2
y = 0
Hence, the system has infinately many solutions (x, y , z) = (2 – z, 0, z) and the friend’s claim is incorrect.

Question 2.
Which of the following samples are biased? If the sample is biased, explain why it is biased.
A. A restaurant asks customers to participate in a survey about the food sold at the restaurant. The restaurant uses the surveys that are returned.
Answer:
The sample is biased because only customers with strong opinions will return the survey

B. You want to know the favorite sport of students at your school. You randomly select athletes to survey at the winter sports banquet.
Answer:
The sample is biased because the survey doesn’t participate randomly selected students.

C. The owner of a store wants to know whether the store should stay open 1 hour later each night. Each cashier surveys every fifth customer.
Answer:
The sample is nonbiased because in the survey participants randomly selected customers.

D. The owner of a movie theater wants to know whether the volume of its movies is too loud. Patrons under the age of 18 are randomly surveyed.
Answer:
The sample is biased because only patrons under the age of 18 participate in the survey.

Question 3.
A survey asks adults about their favorite way to eat ice cream. The results of the survey are displayed in the table shown.

a. How many people were surveyed?
b. Why might the conclusion, “Adults generally do not prefer to eat their ice cream in a cone” be inaccurate to draw from this data?
c. You decide to test the results of the poll by surveying adults chosen at random. What is the probability that at least three out of the six people you survey prefer to eat ice cream in a cone?
d. Four of the six respondents in your study said they prefer to eat their ice cream in a cone. You conclude that the other survey is inaccurate. Why might this conclusion be incorrect?
e. What is the margin of error for your survey?
Answer:
a. 2246
b. The conclusion is inaccurate
c. 0.2374
d. Conclusion is wrong
e. ±40.82

Explanation:
a. Given that,
Margin error = ±2.11
±0.0211 = ±\(\frac { 1 }{ √n } \)
(±0.0211)² = (±\(\frac { 1 }{ √n } \))²
0.000445 = \(\frac { 1 }{ n } \)
n = 2246
b. From the result if the survey we see that people prefer different types of ice cream with different percentages and as 2915 people prefer ice cream in a cone, we cannot conclude that in general people do not eat ice cream in a cone.
c. from the result of the survey we see that 29%of people prefer ice cream in acone. So we can compute the probability that a person chooses ice cream in the cone and the probability that doesn’t choose ice cream in the cone
A: “A person prefers to eat ice cream in a cone”
B: “A person doesn’t prefer to eat ice cream in a cone”
P(A) = \(\frac { 29 }{ 100 } \) = 0.291
P(B) = 1 – P(A) = 1 – 0.29 = 0.71
P(X≥3) = 1 – P(X<3)
= 1 – P(X = 2) – P(x = 1) – P(X = 0)
P(X = 2) = 0.3206
P(X = 1) = 0.3139
P(X = 0) = 0.1281
P(X≥3) = 1 – 0.3206 – 0.3139 – 0.1281
= 0.2374
d. The conclusion might be wrong because in your survey, we have only 6 persons that participate and since the sample size is too small, we cannot conclude that the other survey is inaccurate.
e. The sample size of our survey is 6
The margin error = ±\(\frac { 1 }{ √n } \)
= ±\(\frac { 1 }{ √6 } \)
= ±0.4082
The margin error is ±40.82%

Question 4.
You are making a lampshade out of fabric for the lamp shown. The pattern for the lampshade is shown in the diagram on the left.
a. Use the smaller sector to write an equation that relates θ and x.
b. Use the larger sector to write an equation that relates θ and x+ 10.
c. Solve the system of equations from parts (a) and (b) for x and θ.
d. Find the amount of fabric (in square inches) that you will use to make the lampshade.

Answer:
a. θ = \(\frac { 5π }{ x } \)
b. θ = \(\frac { 14π }{ x + 10} \)
c. x = 5.56, θ = 2.82
d. 298.17

Explanation:
a. r = x, s = 5π
Arc length s = rθ
5π = x . θ
θ = \(\frac { 5π }{ x } \)
b. r = x + 10, s = 14π
Arc length s = rθ
14π = (x + 10)θ
θ = \(\frac { 14π }{ x + 10 } \)
c. From a and b
θ = \(\frac { 5π }{ x } \)
θ = \(\frac { 14π }{ x + 10 } \)
\(\frac { 5π }{ x } \) = \(\frac { 14π }{ x + 10 } \)
5(x + 10) = 14
5x + 50 = 14
x = 5.56
θ = \(\frac { 5π }{ 5.56 } \)
= 2.82 radians
d. Area = \(\frac { 1 }{ 2 } \)r²θ
A<G> = \(\frac { 1 }{ 2 } \)(5.56 + 10)² . 2.82
= \(\frac { 1 }{ 2 } \) . 242.11 . 2.82
= 341.76
A<S> = \(\frac { 1 }{ 2 } \)r²θ
= \(\frac { 1 }{ 2 } \) . (5.56)² . 2.82
= \(\frac { 1 }{ 2 } \) . 30.91 . 2.82
= 43.59
A = A<G> – A<S>
= 341.76 – 43.59
= 298.17

Question 5.
For all students taking the Medical College Admission Test over a period of 3 years, the mean score was 25.1. During the same 3 years, a group of 1000 students who took the test had a mean score of 25.3. Classify each mean as a parameter or a statistic. Explain.
Answer:
25.1 Parameter
25.3 statistic

Explanation:
The mean score of 25.1 is computed from the all students from the college. So, it is a parameter and the mean score of 25.3 is computed from a sample of 1000 students. So, it is a statistic.

Question 6.
Complete the table for the four equations. Explain your reasoning.

Answer:

Question 7.
The normal distribution shown has mean 63 and standard deviation 8. Find the percent of the area under the normal curve that is represented by the shaded region. Then describe another interval underthe normal curve that has the same area.

Answer:
P(55≤x≤83) = P(\(\frac { 55 – 63 }{ 8 } \) ≤\(\frac { x – 63 }{ 8 } \) ≤\(\frac { 83 – 63 }{ 8 } \))
= P(-1 ≤z≤2.5) = P(z<2.5) – P(z<-1)
= 0.9938 – 0.1587 = 0.84
The probability is 0.84 or 84%

Question 8.
Which of the rational expressions cannot be simplified? (HSA-APR.D.6)

Answer:
A. Factors of 2x² + 5x – 3 = 0
= \(\frac { -5  ± √(25 – 4 . 2 . (-3))}{ 2 . 2 } \)
= \(\frac { -5  ± √(25 + 24)}{ 4 } \)
= \(\frac { -5  ± 7}{ 4 } \)
The solutions are \(\frac { 1 }{ 2 } \), -3
2x² + 5x – 3 = 2(x – \(\frac { 1 }{ 2 } \)) (x + 3)
Factors of x² – 7x + 12 = 0
= \(\frac { -(-7)  ± √(49 – 4 . 1 . 12)}{ 2 . 1 } \)
= \(\frac { 7  ± √(49 – 48)}{ 2 } \)
= \(\frac { 7 ± 1}{ 2 } \)
The solutions are 4, 3
x² – 7x + 12 = (x – 4)(x – 3)
\(\frac { 2x² + 5x – 3 }{ x² – 7x + 12 } \)= \(\frac { 2(x – 1/2)(x + 3) }{ (x – 4)(x – 3) } \)