Precalculus

Robert Blitzer

Chapter 3

Exponential and Logarithmic Functions - all with Video Answers

Educators

FG

Section 1

Exponential Functions

Problem 1

In Exercises 1–10 perform the indicated operations and write the result in standard form.
$$
(8-3 i)-(17-7 i)
$$

Thomas Emment

Thomas Emment

Numerade Educator

Problem 2

In Exercises 1–10 perform the indicated operations and write the result in standard form.
$$
4 i(3 i-2)
$$

Thomas Emment

Numerade Educator

Problem 3

In Exercises 1–10 perform the indicated operations and write the result in standard form.
$$
(7-i)(2+3 i)
$$

Thomas Emment

Numerade Educator

Problem 4

In Exercises 1–10 perform the indicated operations and write the result in standard form.
$$
(3-4 i)^{2}
$$

Thomas Emment

Numerade Educator

Problem 5

In Exercises 1–10 perform the indicated operations and write the result in standard form.
$$
(7+8 i)(7-8 i)
$$

Thomas Emment

Numerade Educator

Problem 6

In Exercises 1–10 perform the indicated operations and write the result in standard form.
$$
\frac{6}{5+i}
$$

Thomas Emment

Numerade Educator

Problem 7

In Exercises 1–10 perform the indicated operations and write the result in standard form.
$$
\frac{3+4 i}{4-2 i}
$$

Thomas Emment

Numerade Educator

Problem 8

In Exercises 1–10 perform the indicated operations and write the result in standard form.
$$
\sqrt{-32}-\sqrt{-18}
$$

Thomas Emment

Numerade Educator

Problem 9

In Exercises 1–10 perform the indicated operations and write the result in standard form.
$$
(-2+\sqrt{-100})^{2}
$$

Thomas Emment

Numerade Educator

Problem 10

In Exercises 1–10 perform the indicated operations and write the result in standard form.
$$
\frac{4+\sqrt{-8}}{2}
$$

Thomas Emment

Numerade Educator

Problem 11

In Exercises 11–12, solve each quadratic equation using the quadratic formula. Express solutions in standard form.
$$
x^{2}-2 x+4=0
$$

Thomas Emment

Numerade Educator

Problem 12

In Exercises 11–12, solve each quadratic equation using the quadratic formula. Express solutions in standard form.
$$
2 x^{2}-6 x+5=0
$$

Thomas Emment

Numerade Educator

Problem 13

In Exercises 13–16, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range.
$$
f(x)=-(x+1)^{2}+4
$$

Thomas Emment

Numerade Educator

Problem 14

In Exercises 13–16, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range.
$$
f(x)=(x+4)^{2}-2
$$

Thomas Emment

Numerade Educator

Problem 15

In Exercises 13–16, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range.
$$
f(x)=-x^{2}+2 x+3
$$

Thomas Emment

Numerade Educator

Problem 16

In Exercises 13–16, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range.
$$
f(x)=2 x^{2}-4 x-6
$$

Thomas Emment

Numerade Educator

Problem 17

In Exercises 17–18, use the function’s equation, and not its graph, to find
a. the minimum or maximum value and where it occurs.
b. the function’s domain and its range.
$$
f(x)=-x^{2}+14 x-106
$$

Thomas Emment

Numerade Educator

Problem 18

In Exercises 17–18, use the function’s equation, and not its graph, to find
a. the minimum or maximum value and where it occurs.
b. the function’s domain and its range.
$$
f(x)=2 x^{2}+12 x+703
$$

Thomas Emment

Numerade Educator

Problem 19

A quarterback tosses a football to a receiver 40 yards downfield. The height of the football, f(x), in feet, can be modeled by
$$
f(x)=-0.025 x^{2}+x+6
$$
where x is the ball’s horizontal distance, in yards, from the quarterback.
a. What is the ball’s maximum height and how far from the quarterback does this occur?
b. From what height did the quarterback toss the football?
c. If the football is not blocked by a defensive player nor caught by the receiver, how far down the field will it go before hitting the ground? Round to the nearest tenth of a yard.
d. Graph the function that models the football’s parabolic path.

Sheryl Ezze

Numerade Educator

Problem 20

A field bordering a straight stream is to be enclosed. The side bordering the stream is not to be fenced. If 1000 yards of fencing material is to be used, what are the dimensions of the largest rectangular field that can be fenced? What is the maximum area?

Thomas Emment

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Problem 21

Among all pairs of numbers whose difference is 14, fi nd a pair whose product is as small as possible. What is the minimum product?

Thomas Emment

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Problem 22

You have 1000 feet of fencing to construct six corrals, as shown in the figure. Find the dimensions that maximize the enclosed area. What is the maximum area?

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 23

The annual yield per fruit tree is fairly constant at 150 pounds per tree when the number of trees per acre is 35 or fewer. For each additional tree over 35, the annual yield per tree for all trees on the acre decreases by 4 pounds due to overcrowding. How many fruit trees should be planted per acre to maximize the annual yield for the acre? What is the maximum number of pounds of fruit per acre?

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Problem 24

In Exercises 24–27, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d). ]
$$
f(x)=-x^{3}+x^{2}+2 x
$$

Thomas Emment

Numerade Educator

Problem 25

In Exercises 24–27, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d). ]
$$
f(x)=x^{6}-6 x^{4}+9 x^{2}
$$

Thomas Emment

Numerade Educator

Problem 26

In Exercises 24–27, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d). ]
$$
f(x)=x^{5}-5 x^{3}+4 x
$$

Thomas Emment

Numerade Educator

Problem 27

In Exercises 24–27, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d). ]
$$
f(x)=-x^{4}+1
$$

Thomas Emment

Numerade Educator

Problem 28

The Brazilian Amazon rain forest is the world's largest tropical rain forest, with some of the greatest biodiversity of any region. In 2009 , the number of trees cut down in the Amazon dropped to its lowest level in 20 years. The line graph shows the number of square kilometers cleared from 2001 through 2009
a. Use the Leading Coefficient Test to determine the end behavior to the right for the graph of f .
b. Assume that the rate at which the Amazon rain forest is being cut down continues to decline. Based on your answer to part (a), will f be useful in modeling Amazon deforestation over an extended period of time? Explain your answer.
c. Use the Leading Coefficient Test to determine the end behavior to the right for the graph of g
d. Assume that the rate at which the Amazon rain forest is being cut down continues to decline. Based on your answer to part (c), will g be useful in modeling Amazon deforestation over an extended period of time? Explain your answer. .

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Problem 29

The polynomial function
$$
f(x)=-0.87 x^{3}+0.35 x^{2}+81.62 x+7684.94
$$
model the number of thefts, f(x), in thousands in the United States x years after 1987. Will this function be useful in modeling the number of thefts over an extended period of time? Explain your answer.

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Problem 30

In Exercises 30–31, find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero.
$$
f(x)=-2(x-1)(x+2)^{2}(x+5)^{3}
$$

Thomas Emment

Numerade Educator

Problem 31

In Exercises 30–31, find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero.
$$
f(x)=x^{3}-5 x^{2}-25 x+125
$$

Thomas Emment

Numerade Educator

Problem 32

Show that $f(x)=x^{3}-2 x-1$ has a real zero between 1 and $2 .$

Thomas Emment

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Problem 33

In Exercises 33–38,
a. Use the Leading Coefficient Test to determine the graph’s end behavior.
b. Determine whether the graph has y@axis symmetry, origin symmetry, or neither.
c. Graph the function.
$$
f(x)=x^{3}-x^{2}-9 x+9
$$

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Problem 34

In Exercises 33–38,
a. Use the Leading Coefficient Test to determine the graph’s end behavior.
b. Determine whether the graph has y@axis symmetry, origin symmetry, or neither.
c. Graph the function.
$$
f(x)=4 x-x^{3}
$$

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Problem 35

In Exercises 33–38,
a. Use the Leading Coefficient Test to determine the graph’s end behavior.
b. Determine whether the graph has y@axis symmetry, origin symmetry, or neither.
c. Graph the function.
$$
f(x)=2 x^{3}+3 x^{2}-8 x-12
$$

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Problem 36

In Exercises 33–38,
a. Use the Leading Coefficient Test to determine the graph’s end behavior.
b. Determine whether the graph has y@axis symmetry, origin symmetry, or neither.
c. Graph the function.
$$
f(x)=-x^{4}+25 x^{2}
$$

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Problem 37

In Exercises 33–38,
a. Use the Leading Coefficient Test to determine the graph’s end behavior.
b. Determine whether the graph has y@axis symmetry, origin symmetry, or neither.
c. Graph the function.
$$
f(x)=-x^{4}+6 x^{3}-9 x^{2}
$$

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Problem 38

In Exercises 33–38,
a. Use the Leading Coefficient Test to determine the graph’s end behavior.
b. Determine whether the graph has y@axis symmetry, origin symmetry, or neither.
c. Graph the function.
$$
f(x)=3 x^{4}-15 x^{3}
$$

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Problem 39

In Exercises 39–40, graph each polynomial function.
$$
f(x)=2 x^{2}(x-1)^{3}(x+2)
$$

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Problem 40

In Exercises 39–40, graph each polynomial function.
$$
f(x)=-x^{3}(x+4)^{2}(x-1)
$$

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Problem 41

In Exercises 41–43, divide using long division.
$$
\left(4 x^{3}-3 x^{2}-2 x+1\right) \div(x+1)
$$

Thomas Emment

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Problem 42

In Exercises 41–43, divide using long division.
$$
\left(10 x^{3}-26 x^{2}+17 x-13\right) \div(5 x-3)
$$

Thomas Emment

Numerade Educator

Problem 43

In Exercises 41–43, divide using long division.
$$
\left(4 x^{4}+6 x^{3}+3 x-1\right) \div\left(2 x^{2}+1\right)
$$

Thomas Emment

Numerade Educator

Problem 44

In Exercises 44–45, divide using synthetic division.
$$
\left(3 x^{4}+11 x^{3}-20 x^{2}+7 x+35\right) \div(x+5)
$$

Thomas Emment

Numerade Educator

Problem 45

In Exercises 44–45, divide using synthetic division.
$$
\left(3 x^{4}-2 x^{2}-10 x\right) \div(x-2)
$$

Thomas Emment

Numerade Educator

Problem 46

Given $f(x)=2 x^{3}-7 x^{2}+9 x-3,$ use the Remainder Theorem to find $f(-13) .$

Thomas Emment

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Problem 47

Use synthetic division to divide $f(x)=2 x^{3}+x^{2}-13 x+6$ by $x-2 .$ Use the result to find all zeros of $f .$

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Problem 48

Solve the equation $x^{3}-17 x+4=0$ given that 4 is a root.

Thomas Emment

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Problem 49

In Exercises 49–50, use the Rational Zero Theorem to list all possible rational zeros for each given function.
$$
f(x)=x^{4}-6 x^{3}+14 x^{2}-14 x+5
$$

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Problem 50

In Exercises 49–50, use the Rational Zero Theorem to list all possible rational zeros for each given function.
$$
f(x)=3 x^{5}-2 x^{4}-15 x^{3}+10 x^{2}+12 x-8
$$

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Problem 51

In Exercises 51–52, use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros for each given function.
$$f(x)=3 x^{4}-2 x^{3}-8 x+5$$

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Problem 52

In Exercises 51–52, use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros for each given function.
$$
f(x)=2 x^{5}-3 x^{3}-5 x^{2}+3 x-1
$$

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Problem 53

Use Descartes's Rule of Signs to explain why $2 x^{4}+6 x^{2}+8=0$ has no real roots.

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Problem 54

For Exercises 54–60,
a. List all possible rational roots or rational zeros.
b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros.
c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero.
d. Use the quotient from part (c) to find all the remaining roots or zeros.
$$
f(x)=x^{3}+3 x^{2}-4
$$

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Problem 55

For Exercises 54–60,
a. List all possible rational roots or rational zeros.
b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros.
c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero.
d. Use the quotient from part (c) to find all the remaining roots or zeros.
$$
f(x)=6 x^{3}+x^{2}-4 x+1
$$

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Problem 56

For Exercises 54–60,
a. List all possible rational roots or rational zeros.
b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros.
c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero.
d. Use the quotient from part (c) to find all the remaining roots or zeros.
$$
8 x^{3}-36 x^{2}+46 x-15=0
$$

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Problem 57

For Exercises 54–60,
a. List all possible rational roots or rational zeros.
b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros.
c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero.
d. Use the quotient from part (c) to find all the remaining roots or zeros.
$$
2 x^{3}+9 x^{2}-7 x+1=0
$$

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Problem 58

For Exercises 54–60,
a. List all possible rational roots or rational zeros.
b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros.
c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero.
d. Use the quotient from part (c) to find all the remaining roots or zeros.
$$
x^{4}-x^{3}-7 x^{2}+x+6=0
$$

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Problem 59

For Exercises 54–60,
a. List all possible rational roots or rational zeros.
b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros.
c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero.
d. Use the quotient from part (c) to find all the remaining roots or zeros.
$$
4 x^{4}+7 x^{2}-2=0
$$

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Problem 60

For Exercises 54–60,
a. List all possible rational roots or rational zeros.
b. Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros.
c. Use synthetic division to test the possible rational roots or zeros and find an actual root or zero.
d. Use the quotient from part (c) to find all the remaining roots or zeros.
$$
f(x)=2 x^{4}+x^{3}-9 x^{2}-4 x+4
$$

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Problem 61

In Exercises 61–62, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, graph the function and verify the real zeros and the given function value.
$$
n=3 ; 2 \text { and } 2-3 i \text { are zeros; } f(1)=-10
$$

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Problem 62

In Exercises 61–62, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, graph the function and verify the real zeros and the given function value.
$$
n=4 ; i \text { is a zero; }-3 \text { is a zero of multiplicity } 2 ; f(-1)=16
$$

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Problem 63

In Exercises 63–64, find all the zeros of each polynomial function and write the polynomial as a product of linear factors.
$$
f(x)=2 x^{4}+3 x^{3}+3 x-2
$$

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Problem 64

In Exercises 63–64, find all the zeros of each polynomial function and write the polynomial as a product of linear factors.
$$
g(x)=x^{4}-6 x^{3}+x^{2}+24 x+16
$$

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Problem 65

In Exercises 65–68, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

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Problem 66

In Exercises 65–68, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

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Problem 67

In Exercises 65–68, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

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Problem 68

In Exercises 65–68, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

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Problem 69

In Exercises $69-70,$ use transformations of $f(x)=\frac{1}{x}$ or $f(x)=\frac{1}{x^{2}}$ to graph each rational function.
$$
g(x)=\frac{1}{(x+2)^{2}}-1
$$

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Problem 70

In Exercises $69-70,$ use transformations of $f(x)=\frac{1}{x}$ or $f(x)=\frac{1}{x^{2}}$ to graph each rational function.
$$
h(x)=\frac{1}{x-1}+3
$$

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Problem 71

In Exercises 71–78, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function.
$$
f(x)=\frac{2 x}{x^{2}-9}
$$

Thomas Emment

Numerade Educator

Problem 72

In Exercises 71–78, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function.
$$
g(x)=\frac{2 x-4}{x+3}
$$

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Problem 73

In Exercises 71–78, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function.
$$
h(x)=\frac{x^{2}-3 x-4}{x^{2}-x-6}
$$

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Problem 74

In Exercises 71–78, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function.
$$
r(x)=\frac{x^{2}+4 x+3}{(x+2)^{2}}
$$

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Problem 75

In Exercises 71–78, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function.
$$
y=\frac{x^{2}}{x+1}
$$

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Problem 76

In Exercises 71–78, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function.
$$
y=\frac{x^{2}+2 x-3}{x-3}
$$

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Problem 77

In Exercises 71–78, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function.
$$
f(x)=\frac{-2 x^{3}}{x^{2}+1}
$$

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Problem 78

In Exercises 71–78, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function.
$$
g(x)=\frac{4 x^{2}-16 x+16}{2 x-3}
$$

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Problem 79

A company is planning to manufacture affordable graphing calculators. The fixed monthly cost will be $50,000 and it will cost $25 to produce each calculator.
a. Write the cost function, $C,$ of producing $x$ graphing calculators.
b. Write the average cost function, $\overline{C},$ of producing $x$ graphing calculators.
c. Find and interpret $\overline{C}(50), \quad \overline{C}(100), \quad \overline{C}(1000),$ and
$\overline{C}(100,000)$ .
d. What is the horizontal asymptote for the graph of this function and what does it represent?

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Problem 80

Exercises $80-81$ involve rational functions that model the given situations. In each case, find the horizontal asymptote as $x \rightarrow \infty$ and then describe what this means in practical terms.
$f(x)=\frac{150 x+120}{0.05 x+1} ;$ the number of bass, $f(x),$ after $x$ months in a lake that was stocked with 120 bass

Thomas Emment

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Problem 81

Exercises $80-81$ involve rational functions that model the given situations. In each case, find the horizontal asymptote as $x \rightarrow \infty$ and then describe what this means in practical terms.
$P(x)=\frac{72,900}{100 x^{2}+729} ;$ the percentage, $P(x),$ of people in the United States with $x$ years of education who are unemployed

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Problem 82

The bar graph shows the population of the United States, in millions, for six selected years.
a. Write a function that models the total U.S. population, P(x), in millions, x years after 1985.
b. Write a rational function that models the fraction of men in the U.S. population, R(x), x years after 1985.
c. What is the equation of the horizontal asymptote associated with the function in part (b)? Round to two decimal places. What does this mean about the percentage of men in the U.S. population over time?

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Problem 83

A jogger ran 4 miles and then walked 2 miles. The average velocity running was 3 miles per hour faster than the average velocity walking. Express the total time for running and walking, T, as a function of the average velocity walking, x.

Thomas Emment

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Problem 84

The area of a rectangular floor is 1000 square feet. Express the perimeter of the floor, P, as a function of the width of the rectangle, x.

Thomas Emment

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Problem 85

In Exercises 85–90, solve each inequality and graph the solution set on a real number line.
$$
2 x^{2}+5 x-3<0
$$

Thomas Emment

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Problem 86

In Exercises 85–90, solve each inequality and graph the solution set on a real number line.
$$
2 x^{2}+9 x+4 \geq 0
$$

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Problem 87

In Exercises 85–90, solve each inequality and graph the solution set on a real number line.
$$
\frac{x-6}{x+2}>0
$$

Thomas Emment

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Problem 88

In Exercises 85–90, solve each inequality and graph the solution set on a real number line.
$$
\frac{x-6}{x+2}>0
$$

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Numerade Educator

Problem 89

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If $f(x)=2^{x},$ then $f(a+b)=f(a)+f(b)$

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Problem 90

In Exercises 85–90, solve each inequality and graph the solution set on a real number line.
$$
\frac{x+3}{x-4} \leq 5
$$

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Problem 91

The graph shows stopping distances for motorcycles at various speeds on dry roads and on wet roads.
model a motorcycle’s stopping distance, f(x) or g(x), in feet, traveling at x miles per hour. Function f models stopping distance on dry pavement and function g models stopping distance on wet pavement.
a. Use function g to find the stopping distance on wet pavement for a motorcycle traveling at 35 miles per hour. Round to the nearest foot. Does your rounded answer overestimate or underestimate the stopping distance shown by the graph? By how many feet?
b. Use function f to determine speeds on dry pavement
requiring stopping distances that exceed 267 feet.

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Problem 92

Use the position function
$$
s(t)=-16 t^{2}+v_{0} t+s_{0}
$$
to solve this problem. A projectile is fi red vertically upward from ground level with an initial velocity of 48 feet per second. During which time period will the projectile’s height exceed 32 feet?

Thomas Emment

Numerade Educator

Problem 93

Solve the variation problems in Exercises 93–98.
Many areas of Northern California depend on the snowpack of the Sierra Nevada mountain range for their water supply. The volume of water produced from melting snow varies directly as the volume of snow. Meteorologists have determined that 250 cubic centimeters of snow will melt to 28 cubic centimeters of water. How much water does 1200 cubic centimeters of melting snow produce?

Thomas Emment

Numerade Educator

Problem 94

Solve the variation problems in Exercises 93–98.
The distance that a body falls from rest is directly proportional to the square of the time of the fall. If skydivers fall 144 feet in 3 seconds, how far will they fall in 10 seconds?

Thomas Emment

Numerade Educator

Problem 95

Solve the variation problems in Exercises 93–98.
The pitch of a musical tone varies inversely as its wavelength. A tone has a pitch of 660 vibrations per second and a wavelength of 1.6 feet. What is the pitch of a tone that has a wavelength of 2.4 feet?

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Problem 96

Solve the variation problems in Exercises 93–98.
The loudness of a stereo speaker, measured in decibels, varies inversely as the square of your distance from the speaker. When you are 8 feet from the speaker, the loudness is 28 decibels. What is the loudness when you are 4 feet from the speaker?

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Problem 97

Solve the variation problems in Exercises 93–98.
The time required to assemble computers varies directly as the number of computers assembled and inversely as the number of workers. If 30 computers can be assembled by 6 workers in 10 hours, how long would it take 5 workers to assemble 40 computers?

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Problem 98

Solve the variation problems in Exercises 93–98.
The volume of a pyramid varies jointly as its height and the area of its base. A pyramid with a height of 15 feet and a base with an area of 35 square feet has a volume of 175 cubic feet. Find the volume of a pyramid with a height of 20 feet and a base with an area of 120 square feet.

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Problem 99

Heart rates and life spans of most mammals can be modeled using inverse variation. The bar graph shows the average heart rate and the average life span of five mammals.
a. A mammal’s average life span, L, in years, varies inversely as its average heart rate, R,in beats per minute. Use the data shown for horses to write the equation that models this relationship.
b. Is the inverse variation equation in part (a) an exact model or an approximate model for the data shown for lions?
c. Elephants have an average heart rate of 27 beats per minute. Determine their average life span.

Sheryl Ezze

Numerade Educator