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Calculus for Business, Economics, Life Sciences, and Social Sciences 13th

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Chapter 1

Functions and Graphs

Educators


Problem 1

Use point-by-point plotting to sketch the graph of each equation.
$$
y=x+1
$$

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Problem 2

Use point-by-point plotting to sketch the graph of each equation.
$$
x=y+1
$$

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Problem 3

Use point-by-point plotting to sketch the graph of each equation.
$$
x=y^{2}
$$

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Problem 4

Use point-by-point plotting to sketch the graph of each equation.
$$
y=x^{2}
$$

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Problem 5

Use point-by-point plotting to sketch the graph of each equation.
$$
y=x^{3}
$$

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Problem 6

Use point-by-point plotting to sketch the graph of each equation.
$$
x=y^{3}
$$

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Problem 7

Use point-by-point plotting to sketch the graph of each equation.
$$
x y=-6
$$

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Problem 8

Use point-by-point plotting to sketch the graph of each equation.
$$
x y=12
$$

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Problem 9

Indicate whether each table specifies a function.

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Problem 10

Indicate whether each table specifies a function.

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Problem 11

Indicate whether each table specifies a function.

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Problem 12

Indicate whether each table specifies a function.

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Problem 13

Indicate whether each table specifies a function.

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Problem 14

Indicate whether each table specifies a function.

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Problem 15

Indicate whether each graph specifies a function.

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Problem 16

Indicate whether each graph specifies a function.

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Problem 18

Indicate whether each graph specifies a function.

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Problem 19

Indicate whether each graph specifies a function.

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Problem 20

Indicate whether each graph specifies a function.

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Problem 22

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.
$$
y=10-3 x
$$

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Problem 23

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.
$$
x y-4=0
$$

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Problem 24

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.
$$
x^{2}-y=8
$$

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Problem 25

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.
$$
y=5 x+\frac{1}{2}(7-10 x)
$$

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Problem 26

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.
$$
y=\frac{2+x}{3}+\frac{2-x}{3}
$$

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Problem 27

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.
$$
3 x+4 y=5
$$

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Problem 28

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.
$$
9 x-2 y+6=0
$$

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Problem 29

Use point-by-point plotting to sketch the graph of each function.
$$
f(x)=1-x
$$

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Problem 30

Use point-by-point plotting to sketch the graph of each function.
$$
f(x)=\frac{x}{2}-3
$$

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Problem 31

Use point-by-point plotting to sketch the graph of each function.
$$
f(x)=x^{2}-1
$$

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Problem 32

Use point-by-point plotting to sketch the graph of each function.
$$
f(x)=3-x^{2}
$$

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Problem 33

Use point-by-point plotting to sketch the graph of each function.
$$
f(x)=4-x^{3}
$$

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Problem 34

Use point-by-point plotting to sketch the graph of each function.
$$
f(x)=x^{3}-2
$$

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Problem 35

Use point-by-point plotting to sketch the graph of each function.
$$
f(x)=\frac{8}{x}
$$

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Problem 36

Use point-by-point plotting to sketch the graph of each function.
$$
f(x)=\frac{-6}{x}
$$

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Problem 37

The three points in the table are on the graph of the indicated function $f .$ Do these three points provide sufficient information for you to sketch the graph of $y=f(x)$ ? Add more points to the table until you are satisfied that your sketch is a good representation of the graph of $y=f(x)$ for $-5 \leq x \leq 5$.
$$
\begin{array}{lccc}
x & -1 & 0 & 1 \\
f(x) & -1 & 0 & 1
\end{array} \quad f(x)=\frac{2 x}{x^{2}+1}
$$

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Problem 38

The three points in the table are on the graph of the indicated function $f .$ Do these three points provide sufficient information for you to sketch the graph of $y=f(x)$ ? Add more points to the table until you are satisfied that your sketch is a good representation of the graph of $y=f(x)$ for $-5 \leq x \leq 5$.
$$
\begin{array}{lccc}
x & 0 & 1 & 2 \\
f(x) & 0 & 1 & 2
\end{array} \quad f(x)=\frac{3 x^{2}}{x^{2}+2}
$$

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Problem 39

Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some
problems may have more than one answer.
$$
y=f(-5)
$$

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Problem 40

Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some
problems may have more than one answer.
$$
y=f(4)
$$

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Problem 41

Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some
problems may have more than one answer.
$$
y=f(5)
$$

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Problem 42

Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some
problems may have more than one answer.
$$
y=f(-2)
$$

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Problem 43

Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some
problems may have more than one answer.
$$
0=f(x)
$$

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Problem 44

Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some
problems may have more than one answer.
$$
3=f(x), x<0
$$

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Problem 45

Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some
problems may have more than one answer.
$$
-4=f(x)
$$

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Problem 46

Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some
problems may have more than one answer.
$$
4=f(x)
$$

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Problem 47

Find the domain of each function.
$$
F(x)=2 x^{3}-x^{2}+3
$$

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Problem 48

Find the domain of each function.
$$
H(x)=7-2 x^{2}-x^{4}
$$

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Problem 49

Find the domain of each function.
$$
f(x)=\frac{x-2}{x+4}
$$

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Problem 50

Find the domain of each function.
$$
g(x)=\frac{x+1}{x-2}
$$

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Problem 51

Find the domain of each function.
$$
g(x)=\sqrt{7-x}
$$

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Problem 52

Find the domain of each function.
$$
F(x)=\frac{1}{\sqrt{5+x}}
$$

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Problem 53

Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.
$$
2 x+5 y=10
$$

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Problem 54

Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.
$$
6 x-7 y=21
$$

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Problem 55

Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.
$$
y(x+y)=4
$$

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Problem 56

Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.
$$
x(x+y)=4
$$

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Problem 57

Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.
$$
x^{-3}+y^{3}=27
$$

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Problem 58

Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.
$$
x^{2}+y^{2}=9
$$

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Problem 59

Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.
$$
x^{3}-y^{2}=0
$$

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Problem 60

Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.
$$
\sqrt{x}-y^{3}=0
$$

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Problem 61

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(4)
$$

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Problem 62

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(-5)
$$

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Problem 63

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(x+1)
$$

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Problem 64

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(x-2)
$$

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Problem 65

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(-6 x)
$$

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Problem 66

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(10 x)
$$

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Problem 67

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f\left(x^{3}\right)
$$

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Problem 68

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(\sqrt{x})
$$

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Problem 69

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(2)+f(h)
$$

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Problem 70

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(-3)+f(h)
$$

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Problem 71

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(2+h)
$$

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Problem 72

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(-3+h)
$$

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Problem 73

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(2+h)-f(2)
$$

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Problem 74

Find and simplify the expression if $f(x)=x^{2}-4$.
$$
f(-3+h)-f(-3)
$$

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Problem 75

Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$
(A) $f(x+h)$
(B) $f(x+h)-f(x)$
(C) $\frac{f(x+h)-f(x)}{h}$
$$
f(x)=4 x-3
$$

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Problem 76

Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$
(A) $f(x+h)$
(B) $f(x+h)-f(x)$
(C) $\frac{f(x+h)-f(x)}{h}$
$$
f(x)=-3 x+9
$$

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Problem 77

Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$
(A) $f(x+h)$
(B) $f(x+h)-f(x)$
(C) $\frac{f(x+h)-f(x)}{h}$
$$
f(x)=4 x^{2}-7 x+6
$$

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Problem 78

Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$
(A) $f(x+h)$
(B) $f(x+h)-f(x)$
(C) $\frac{f(x+h)-f(x)}{h}$
$$
f(x)=3 x^{2}+5 x-8
$$

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Problem 79

Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$
(A) $f(x+h)$
(B) $f(x+h)-f(x)$
(C) $\frac{f(x+h)-f(x)}{h}$
$$
f(x)=x(20-x)
$$

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Problem 80

Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$
(A) $f(x+h)$
(B) $f(x+h)-f(x)$
(C) $\frac{f(x+h)-f(x)}{h}$
$$
f(x)=x(x+40)
$$

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Problem 81

Refer to the area A and perimeter $P$ of a rectangle with length $l$ and width $w$ (see the figure).
The area of a rectangle is $25 \mathrm{sq}$ in. Express the perimeter $P(w)$ as a function of the width $w,$ and state the domain of this function.

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Problem 82

Refer to the area A and perimeter $P$ of a rectangle with length $l$ and width $w$ (see the figure).
The area of a rectangle is 81 sq in. Express the perimeter $P(l)$ as a function of the length $l,$ and state the domain of this function.

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Problem 83

Refer to the area A and perimeter $P$ of a rectangle with length $l$ and width $w$ (see the figure).
The perimeter of a rectangle is $100 \mathrm{~m}$. Express the area $A(l)$ as a function of the length $l,$ and state the domain of this function.

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Problem 84

Refer to the area A and perimeter $P$ of a rectangle with length $l$ and width $w$ (see the figure).
The perimeter of a rectangle is $160 \mathrm{~m}$. Express the area $A(w)$ as a function of the width $w,$ and state the domain of this function.

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Problem 85

A company manufactures memory chips for microcomputers. Its marketing research department, using statistical techniques, collected the data shown in Table 8 where $p$ is the wholesale price per chip at which $x$ million chips can be sold. Using special analytical techniques (regression analysis), an analyst produced the following price-demand function to model the data:
$$
p(x)=75-3 x \quad 1 \leq x \leq 20
$$
$$
\begin{array}{cc}
\hline x \text { (millions) } & p(\$) \\
1 & 72 \\
4 & 63 \\
9 & 48 \\
14 & 33 \\
20 & 15
\end{array}
$$
(A) Plot the data points in Table 8 , and sketch a graph of the price-demand function in the same coordinate system.
(B) What would be the estimated price per chip for a demand of 7 million chips? For a demand of 11 million chips?

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Problem 86

A company manufactures notebook computers. Its marketing research department, using statistical techniques, collected the data shown in Table $9,$ where $p$ is the wholesale price per computer at which $x$ thousand computers can be sold. Using special analytical techniques (regression analysis), an analyst produced the following price-demand function to model the data:
$$
p(x)=2,000-60 x \quad 1 \leq x \leq 25
$$
$$
\begin{array}{cc}
\hline \boldsymbol{x} \text { (thousands) } & p(\$) \\
1 & 1,940 \\
8 & 1,520 \\
16 & 1,040 \\
21 & 740 \\
25 & 500 \\
\hline
\end{array}
$$
(A) Plot the data points in Table 9 , and sketch a graph of the price-demand function in the same coordinate system.
(B) What would be the estimated price per computer for a demand of 11,000 computers? For a demand of 18,000 computers?

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Problem 87

(A) Using the price-demand function
$$
p(x)=75-3 x \quad 1 \leq x \leq 20
$$
from Problem $85,$ write the company's revenue function and indicate its domain.
(B) Complete Table 10 , computing revenues to the nearest million dollars.
(C) Plot the points from part (B) and sketch a graph of the revenue function using these points. Choose millions for the units on the horizontal and vertical axes.

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Problem 88

(A) Using the price-demand function
$$
p(x)=2,000-60 x \quad 1 \leq x \leq 25
$$
from Problem 86 , write the company's revenue function and indicate its domain.
(B) Complete Table 11 , computing revenues to the nearest thousand dollars.
(C) Plot the points from part (B) and sketch a graph of the revenue function using these points. Choose thousands for the units on the horizontal and vertical axes.

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Problem 89

The financial department for the company in Problems 85 and 87 established the following cost function for producing and selling $x$ million memory chips:
$C(x)=125+16 x$ million dollars
(A) Write a profit function for producing and selling $x$ million memory chips and indicate its domain.
(B) Complete Table 12 , computing profits to the nearest million dollars.
(C) Plot the points in part (B) and sketch a graph of the profit function using these points.

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Problem 90

The financial department for the company in Problems 86 and 88 established the following cost function for producing and selling $x$ thousand notebook computers:
$$
C(x)=4,000+500 x \text { thousand dollars }
$$
(A) Write a profit function for producing and selling $x$ thousand notebook computers and indicate its domain.
(B) Complete Table 13 , computing profits to the nearest thousand dollars.
(C) Plot the points in part (B) and sketch a graph of the profit function using these points.

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Problem 91

A candy box will be made out of a piece of cardboard that measures 8 by 12 in. Equal-sized squares $x$ inches on a side will be cut out of each corner, and then the ends and sides will be folded up to form a rectangular box.
(A) Express the volume of the box $V(x)$ in terms of $x$.
(B) What is the domain of the function $V$ (determined by the physical restrictions)?
(C) Complete Table 14 .
(D) Plot the points in part (C) and sketch a graph of the volume function using these points.

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Problem 92

Refer to Problem $91 .$
(A) Table 15 shows the volume of the box for some values of $x$ between 1 and 2 . Use these values to estimate to one
decimal place the value of $x$ between 1 and 2 that would produce a box with a volume of $65 \mathrm{cu}$ in.
$$
\begin{array}{cc}
\boldsymbol{x} & V(x) \\
1.1 & 62.524 \\
1.2 & 64.512 \\
1.3 & 65.988 \\
1.4 & 66.976 \\
1.5 & 67.5 \\
1.6 & 67.584 \\
1.7 & 67.252 \\
\hline
\end{array}
$$
(B) Describe how you could refine this table to estimate $x$ to two decimal places.
(C) Carry out the refinement you described in part (B) and approximate $x$ to two decimal places.

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Problem 93

Refer to Problems 91 and $92 .$
(A) Examine the graph of $V(x)$ from Problem $91 \mathrm{D}$ and discuss the possible locations of other values of $x$ that would produce a box with a volume of $65 \mathrm{cu}$ in.
(B) Construct a table like Table 15 to estimate any such value to one decimal place.
(C) Refine the table you constructed in part (B) to provide an approximation to two decimal places.

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Problem 94

A parcel delivery service will only deliver packages with length plus girth (distance around) not exceeding 108 in. A rectangular shipping box with square ends $x$ inches on a side is to be used.
(A) If the full 108 in. is to be used, express the volume of the box $V(x)$ in terms of $x$.
(B) What is the domain of the function $V$ (determined by the physical restrictions)?
(C) Complete Table 16 .
(D) Plot the points in part (C) and sketch a graph of the volume function using these points.

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Problem 95

In a study of the speed of muscle contraction in frogs under various loads, British biophysicist
A. W. Hill determined that the weight $w$ (in grams) placed on the muscle and the speed of contraction $v$ (in centimeters per second) are approximately related by an equation of the form
$$
(w+a)(v+b)=c
$$
where $a, b,$ and $c$ are constants. Suppose that for a certain muscle, $a=15, b=1,$ and $c=90 .$ Express $v$ as a function of $w$. Find the speed of contraction if a weight of $16 \mathrm{~g}$ is placed on the muscle.

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Problem 96

The percentage $s$ of seats in the House of Representatives won by Democrats and the percentage $v$ of votes cast for Democrats (when expressed as decimal fractions) are related by the equation
$$
5 v-2 s=1.4 \quad 0<s<1, \quad 0.28<v<0.68
$$
(A) Express $v$ as a function of $s$ and find the percentage of votes required for the Democrats to win $51 \%$ of the seats.
(B) Express $s$ as a function of $v$ and find the percentage of seats won if Democrats receive $51 \%$ of the votes.

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