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Algebra 2

Prentice Hall, Basia Kennedy, Dan Ramirez

Chapter 9

Rational Functions

Educators

CC
DP

Problem 1

Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.
$$
x=1 \text { when } y=11
$$

CC
Charles C.
Numerade Educator

Problem 2

Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.
$$
x=-13 \text { when } y=100
$$

Pranav S.
Numerade Educator

Problem 3

Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.
$$
x=1 \text { when } y=1
$$

CC
Charles C.
Numerade Educator

Problem 4

Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.
$$
x=28 \text { when } y=-2
$$

Pranav S.
Numerade Educator

Problem 5

Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.
$$
x=1.2 \text { when } y=3
$$

CC
Charles C.
Numerade Educator

Problem 6

Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.
$$
x=2.5 \text { when } y=100
$$

Pranav S.
Numerade Educator

Problem 7

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.
$$
\begin{array}{|c|c|c|c|c|}\hline x & {3} & {8} & {10} & {22} \\ \hline y & {15} & {40} & {50} & {110} \\ \hline\end{array}
$$

CC
Charles C.
Numerade Educator

Problem 8

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.
$$
\begin{array}{|c|c|c|c|c|}\hline x & {3} & {5} & {7} & {10.5} \\ \hline y & {14} & {8.4} & {6} & {4} \\ \hline\end{array}
$$

DP
Daniel P.
Numerade Educator

Problem 9

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.
$$
\begin{array}{|c|c|c|c|c|}\hline x & {0.5} & {2.1} & {3.5} & {11} \\ \hline y & {1} & {4.2} & {7} & {22} \\ \hline\end{array}
$$

CC
Charles C.
Numerade Educator

Problem 10

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.
$$
\begin{array}{|c|c|c|c|c|}\hline x & {0.1} & {3} & {6} & {24} \\ \hline y & {3} & {0.1} & {0.05} & {0.0125} \\ \hline\end{array}
$$

DP
Daniel P.
Numerade Educator

Problem 11

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.
$$
\begin{array}{|c|c|c|c|c|}\hline x & {7} & {3} & {1} & {\frac{1}{5}} \\ \hline y & {\frac{1}{7}} & {\frac{1}{3}} & {1} & {5} \\ \hline\end{array}
$$

CC
Charles C.
Numerade Educator

Problem 12

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.
$$
\begin{array}{|c|c|c|c|c|}\hline x & {10} & {12} & {20} & {23} \\ \hline y & {2} & {2 \frac{2}{5}} & {4} & {5 \frac{3}{5}} \\ \hline\end{array}
$$

DP
Daniel P.
Numerade Educator

Problem 13

Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation and find $y$ when $x=10 .$
$$
x=20 \text { when } y=5
$$

CC
Charles C.
Numerade Educator

Problem 14

Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation and find $y$ when $x=10 .$
$$
x=20 \text { when } y=-4
$$

Pranav S.
Numerade Educator

Problem 15

Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation and find $y$ when $x=10 .$
$$
x=5 \text { when } y=-\frac{1}{3}
$$

CC
Charles C.
Numerade Educator

Problem 16

Describe the combined variation that is modeled by each formula.
$$
A=\pi r^{2}
$$

DP
Daniel P.
Numerade Educator

Problem 17

Describe the combined variation that is modeled by each formula.
$$
A=0.5 b h
$$

CC
Charles C.
Numerade Educator

Problem 18

Describe the combined variation that is modeled by each formula.
$$
h=\frac{2 A}{b}
$$

DP
Daniel P.
Numerade Educator

Problem 19

Describe the combined variation that is modeled by each formula.
$$
V=\frac{B h}{3}
$$

CC
Charles C.
Numerade Educator

Problem 20

Describe the combined variation that is modeled by each formula.
$$
V=\pi r^{2} h
$$

Adam D.
Numerade Educator

Problem 21

Describe the combined variation that is modeled by each formula.
$$
h=\frac{V}{\pi r^{2}}
$$

CC
Charles C.
Numerade Educator

Problem 22

Describe the combined variation that is modeled by each formula.
$$
V=\ell w h
$$

Adam D.
Numerade Educator

Problem 23

Describe the combined variation that is modeled by each formula.
$$
\ell=\frac{V}{w h}
$$

CC
Charles C.
Numerade Educator

Problem 24

Write the function that models each variation. Find $z$ when $x=4$ and $y=9$
$z$ varies directly with $x$ and inversely with $y .$ When $x=6$ and $y=2, z=15$

Adam D.
Numerade Educator

Problem 25

Write the function that models each variation. Find $z$ when $x=4$ and $y=9$
$z$ varies jointly with $x$ and $y .$ When $x=2$ and $y=3, z=60$

CC
Charles C.
Numerade Educator

Problem 26

Write the function that models each variation. Find $z$ when $x=4$ and $y=9$
$z$ varies directly with the square of $x$ and inversely with $y .$ When $x=2$ and $y=4, z=3 .$

Adam D.
Numerade Educator

Problem 27

Write the function that models each variation. Find $z$ when $x=4$ and $y=9$
$z$ varies inversely with the product of $x$ and $y .$ When $x=2$ and $y=4, z=0.5$

CC
Charles C.
Numerade Educator

Problem 28

Write the function that models each variation. Find $z$ when $x=4$ and $y=9$
a. The spreadsheet shows data that could be modeled by an equation of the form $P V=k$ . Estimate the value of $k$ .
b. Estimate $P$ when $V=62$ .

Adam D.
Numerade Educator

Problem 29

Each ordered pair is from an inverse variation. Find the constant of variation.
$$
(6,3)
$$

CC
Charles C.
Numerade Educator

Problem 30

Each ordered pair is from an inverse variation. Find the constant of variation.
$$
(0.9,4)
$$

DP
Daniel P.
Numerade Educator

Problem 31

Each ordered pair is from an inverse variation. Find the constant of variation.
$$
\left(\frac{3}{8}, \frac{2}{3}\right)
$$

CC
Charles C.
Numerade Educator

Problem 32

Each ordered pair is from an inverse variation. Find the constant of variation.
$$
(\sqrt{2}, \sqrt{18})
$$

Adam D.
Numerade Educator

Problem 33

Each ordered pair is from an inverse variation. Find the constant of variation.
$$
(\sqrt{3}, \sqrt{27})
$$

CC
Charles C.
Numerade Educator

Problem 34

Each ordered pair is from an inverse variation. Find the constant of variation.
$$
(\sqrt{8}, \sqrt{32})
$$

Adam D.
Numerade Educator

Problem 35

Mechanics Gear A drives Gear B. Gear A has $a$ teeth and speed $r_{A}$ in revolutions per minute $(r p m) .$ Gear $B$ has $b$ teeth and speed $r_{B} .$ The quantities are related by the formula $a r_{\mathrm{A}}=b r_{\mathrm{B}}$ Gear $\mathrm{A}$ has 60 teeth and speed 540 $\mathrm{rpm} .$ Gear $\mathrm{B}$ has 45 teeth. Find the speed of Gear $\mathrm{B}$ .

CC
Charles C.
Numerade Educator

Problem 36

Physics The force $F$ of gravity on a rocket varies directly with its mass $m$ and inversely with the square of its distance $d$ from Earth. Write a model for this combined variation. $k_{d^{2}}^{m}$

Adam D.
Numerade Educator

Problem 37

Each pair of values is from a direct variation. Find the missing value.
$$
(3,7),(8, y)
$$

CC
Charles C.
Numerade Educator

Problem 38

Each pair of values is from a direct variation. Find the missing value.
$$
(2,5),(4, y)
$$

Pranav S.
Numerade Educator

Problem 39

Each pair of values is from a direct variation. Find the missing value.
$$
(4,6),(x, 3)
$$

CC
Charles C.
Numerade Educator

Problem 40

Each pair of values is from a direct variation. Find the missing value.
$$
(9,5),(x, 3)
$$

Pranav S.
Numerade Educator

Problem 41

Each pair of values is from a direct variation. Find the missing value.
$$
(8.3,7.1),(5, y)
$$

CC
Charles C.
Numerade Educator

Problem 42

Each pair of values is from a direct variation. Find the missing value.
$$
(2.6,4.5),(x, 6.3)
$$

Pranav S.
Numerade Educator

Problem 43

Each pair of values is from an inverse variation. Find the missing value.
$$
(3,7),(8, y)
$$

CC
Charles C.
Numerade Educator

Problem 44

Each pair of values is from an inverse variation. Find the missing value.
$$
(2,5),(4, y)
$$

Pranav S.
Numerade Educator

Problem 45

Each pair of values is from an inverse variation. Find the missing value.
$$
(4,6),(x, 3)
$$

CC
Charles C.
Numerade Educator

Problem 46

Each pair of values is from an inverse variation. Find the missing value.
$$
(9,5),(x, 3)
$$

Pranav S.
Numerade Educator

Problem 47

Each pair of values is from an inverse variation. Find the missing value.
$$
(8.3,7.1),(5, y)
$$

CC
Charles C.
Numerade Educator

Problem 48

Each pair of values is from an inverse variation. Find the missing value.
$$
(2.6,4.5),(x, 6.3)
$$

Pranav S.
Numerade Educator

Problem 49

Suppose that $y$ varies inversely with the square of $x,$ and $y=50$ when $x=4$ . Find $y$ when $x=5 .$

CC
Charles C.
Numerade Educator

Problem 50

Suppose that $c$ varies jointly with $d$ and the square of $g,$ and $c=30$ when $d=15$ and $g=2 .$ Find $d$ when $c=6$ and $g=8$ .

Adam D.
Numerade Educator

Problem 51

Suppose that $d$ varies jointly with $r$ and $t,$ and $d=110$ when $r=55$ and $t=2$ . Find $r$ when $d=40$ and $t=3 .$

CC
Charles C.
Numerade Educator

Problem 52

Construction A concrete supplier sells premixed concrete in $300-f t^{3}$ truckloads. The area $A$ that the concrete will cover is inversely proportional to the depth $d$ of the concrete.
a. Write a model for the relationship between the area and the depth of a truckload of poured concrete.
b. What area will the concrete cover if it is poured to a depth of 0.5 $\mathrm{ft}$ ? A depth of 1 $\mathrm{ft}$ ? A depth of 1.5 $\mathrm{ft}$ ?
c. When the concrete is poured into a circular area, the depth of the concrete is inversely proportional to the square of the radius $r$ . Write a model for this relationship.

Adam D.
Numerade Educator

Problem 53

Suppose that $y$ varies directly with $x$ and inversely with $z^{2},$ and $x=48$ when $y=8$ and $z=3 .$ Find $x$ when $y=12$ and $z=2$ .

CC
Charles C.
Numerade Educator

Problem 54

Suppose that $t$ varies directly with $s$ and inversely with the square of $r .$ How is the value of $t$ changed when the value of $s$ is doubled? Is tripled?

Adam D.
Numerade Educator

Problem 55

Suppose that $x$ varies directly with the square of $y$ and inversely with $z .$ How is the value of $x$ changed if the value of $y$ is halved? Is quartered?

CC
Charles C.
Numerade Educator

Problem 56

Writing. Explain why 0 cannot be in the domain of an inverse variation.

DP
Daniel P.
Numerade Educator

Problem 57

Critical Thinking. Suppose that $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ are values from an inverse variation. Show that $\frac{x_{1}}{x_{2}}=\frac{y_{2}}{y_{1}} .$

CC
Charles C.
Numerade Educator

Problem 58

Open-Ended. The height $h$ of a cylinder varies directly with its volume $V$ and inversely with the square of its radius $r .$ Find at least four ways to change the volume and radius of a cylinder so that its height is quadrupled.

Adam D.
Numerade Educator

Problem 59

Health Health care professionals use the body mass index (BMI) to establish guidelines for determining any possible risk of their patients and for planing any useful preventative programs. The BMI varies directly with weight and inversely with the square of height. Use this portion of the BMI chart to determine the BMI formula.

CC
Charles C.
Numerade Educator

Problem 60

Which equation does NOT represent inverse variation between $x$ and $z ?$
$$
\begin{array}{ll}{\text { A. } x=\frac{y}{z}} & {\text { B. } x=\frac{-15 z}{y}} \\ {\text { C. } z=\frac{-15 y}{x}} & {\text { D. } x z=5 y}\end{array}
$$

Adam D.
Numerade Educator

Problem 61

If $p$ and $q$ vary inversely, and $p=10$ when $q=-4,$ what is $q$ when $p=-2 ?$
$$
\begin{array}{lllll}{\text { F. } 20} & {\text { G. } \frac{4}{5}} & {\text { H. }-\frac{4}{5}} & {\text { 1. }-20}\end{array}
$$

CC
Charles C.
Numerade Educator

Problem 62

Which equation shows that $z$ varies directly with the square of $x$ and inversely with the cube of $y ?$
$$
\begin{array}{llll}{\text { A. } z=\frac{x^{2}}{y^{3}}} & {\text { B. } z=\frac{x^{3}}{y^{2}}} & {\text { C. } z=\frac{y^{2}}{x^{3}}} & {\text { D. } z=\frac{y^{3}}{x^{2}}}\end{array}
$$

Adam D.
Numerade Educator

Problem 63

Describe how the variables $A$ and $r$ vary in the formula for the area of a circle, $A=\pi r^{2} .$

CC
Charles C.
Numerade Educator

Problem 64

Which data set shows inverse variation: $(24.4,4.8)$ and $(9.6,12.2),$ or $(24.0,4.5)$ and $(18.0,6.5) ?$ Explain.

Adam D.
Numerade Educator

Problem 65

Solve each equation.
$$
\ln 4+\ln x=5
$$

CC
Charles C.
Numerade Educator

Problem 66

Solve each equation.
$$
\ln x-\ln 3=4
$$

DP
Daniel P.
Numerade Educator

Problem 67

Solve each equation.
$$
2 \ln x+3 \ln 4=4
$$

CC
Charles C.
Numerade Educator

Problem 68

Multiply and simplify. Assume that all variables are positive.
$$
-5 \sqrt{6 x} \cdot 3 \sqrt{6 x^{2}}
$$

DP
Daniel P.
Numerade Educator

Problem 69

Multiply and simplify. Assume that all variables are positive.
$$
3 \sqrt[3]{4 x^{2}} \cdot 7 \sqrt[3]{12 x^{4}}
$$

CC
Charles C.
Numerade Educator

Problem 70

Multiply and simplify. Assume that all variables are positive.
$$
\sqrt{5 x^{3}} \cdot \sqrt{40 x y^{7}}
$$

DP
Daniel P.
Numerade Educator

Problem 71

Simplify each radical expression. Use absolute value bars where they are needed.
$$
\sqrt{x^{10} y^{100}}
$$

CC
Charles C.
Numerade Educator

Problem 72

Simplify each radical expression. Use absolute value bars where they are needed.
$$
\sqrt[3]{-64 a^{3} b^{6}}
$$

DP
Daniel P.
Numerade Educator

Problem 73

Simplify each radical expression. Use absolute value bars where they are needed.
$$
\sqrt[4]{64 m^{8} n^{4}}
$$

CC
Charles C.
Numerade Educator

Problem 74

Simplify each radical expression. Use absolute value bars where they are needed.
$$
\sqrt[4]{x^{4}}
$$

DP
Daniel P.
Numerade Educator