Finite Mathematics and Applied Calculus

Stefan Waner, Steven Costenoble

Chapter 9

Nonlinear Functions and Models - all with Video Answers

Educators

Section 1

Quadratic Functions and Models

Problem 1

Sketch the graphs of the quadratic functions, indicating the coordinates of the vertex, the y-intercept, and the $x$ -intercepts (if any).
$$
f(x)=x^{2}+3 x+2
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 2

Sketch the graphs of the quadratic functions, indicating the coordinates of the vertex, the y-intercept, and the $x$ -intercepts (if any).
$$
f(x)=-x^{2}-x
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 3

Sketch the graphs of the quadratic functions, indicating the coordinates of the vertex, the y-intercept, and the $x$ -intercepts (if any).
$$
f(x)=-x^{2}+4 x-4
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 4

Sketch the graphs of the quadratic functions, indicating the coordinates of the vertex, the y-intercept, and the $x$ -intercepts (if any).
$$
f(x)=x^{2}+2 x+1
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 5

Sketch the graphs of the quadratic functions, indicating the coordinates of the vertex, the y-intercept, and the $x$ -intercepts (if any).
$$
f(x)=-x^{2}-40 x+500
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 6

Sketch the graphs of the quadratic functions, indicating the coordinates of the vertex, the y-intercept, and the $x$ -intercepts (if any).
$$
f(x)=x^{2}-10 x-600
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 7

Sketch the graphs of the quadratic functions, indicating the coordinates of the vertex, the y-intercept, and the $x$ -intercepts (if any).
$$
f(x)=x^{2}+x-1
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 8

Sketch the graphs of the quadratic functions, indicating the coordinates of the vertex, the y-intercept, and the $x$ -intercepts (if any).
$$
f(x)=x^{2}+\sqrt{2} x+1
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 9

Sketch the graphs of the quadratic functions, indicating the coordinates of the vertex, the y-intercept, and the $x$ -intercepts (if any).
$$
f(x)=x^{2}+1
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 10

Sketch the graphs of the quadratic functions, indicating the coordinates of the vertex, the y-intercept, and the $x$ -intercepts (if any).
$$
f(x)=-x^{2}+5
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 11

For each demand equation, express the total revenue $R$ as a function of the price $p$ per item, sketch the graph of the resulting function, and determine the price $p$ that maximizes total revenue in each case.
$$
q=-4 p+100
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 12

For each demand equation, express the total revenue $R$ as a function of the price $p$ per item, sketch the graph of the resulting function, and determine the price $p$ that maximizes total revenue in each case.
$$
q=-3 p+300
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 13

For each demand equation, express the total revenue $R$ as a function of the price $p$ per item, sketch the graph of the resulting function, and determine the price $p$ that maximizes total revenue in each case.
$$
q=-2 p+400
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 14

For each demand equation, express the total revenue $R$ as a function of the price $p$ per item, sketch the graph of the resulting function, and determine the price $p$ that maximizes total revenue in each case.
$$
q=-5 p+1,200
$$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 15

Use technology to find the quadratic regression curve through the given points. (Round all coefficients to four decimal places.)
$$
\{(1,2),(3,5),(4,3),(5,1)\}
$$

Carson Merrill

Numerade Educator

Problem 16

Use technology to find the quadratic regression curve through the given points. (Round all coefficients to four decimal places.)
$$
\{(-1,2),(-3,5),(-4,3),(-5,1)\}
$$

Carson Merrill

Numerade Educator

Problem 17

Use technology to find the quadratic regression curve through the given points. (Round all coefficients to four decimal places.)
$$
\{(-1,2),(-3,5),(-4,3)\}
$$

Carson Merrill

Numerade Educator

Problem 18

Use technology to find the quadratic regression curve through the given points. (Round all coefficients to four decimal places.)
$$
\{(2,5),(3,5),(5,3)\}
$$

Carson Merrill

Numerade Educator

Problem 19

The following chart shows total military and arms trade expenditure from 1990 to $2008(t=0$ represents 1990 ).
a. If you want to model the expenditure figures with a function of the form
$$
f(t)=a t^{2}+b t+c
$$
would you expect the coefficient $a$ to be positive or negative? Why? HINT [See "Features of a Parabola," page 621.]
b. Which of the following models best approximates the data given? (Try to answer this without actually computing values.)
(A) $f(t)=5 t^{2}-80 t-1,150$
(B) $f(t)=-5 t^{2}-80 t+1,150$
(C) $f(t)=5 t^{2}-80 t+1,150$
(D) $f(t)=-5 t^{2}-80 t-1,150$
c. What is the nearest year that would correspond to the vertex of the graph of the correct model from part (b)? What is the danger of extrapolating the data in either direction?

Carson Merrill

Numerade Educator

Problem 20

The following chart shows the percentage of the U.S. Discretionary Budget allocated to education from 2003 to 2009 ( $t=3$ represents the start of 2003 ).
a. If you want to model the percentage figures with a function of the form
$$
f(t)=a t^{2}+b t+c
$$
would you expect the coefficient $a$ to be positive or negative? Why? HINT [See "Features of a Parabola" page 621.]
b. Which of the following models best approximates the data given? (Try to answer this without actually computing values)
(A) $f(t)=0.04 t^{2}+0.3 t-6$
(B) $f(t)=-0.04 t^{2}+0.3 t+6$
(C) $f(t)=0.04 t^{2}+0.3 t+6$
(D) $f(t)=-0.04 t^{2}+0.3 t-6$
c. What is the nearest year that would correspond to the vertex of the graph of the correct model from part (b)? What is the danger of extrapolating the data in either direction?

Carson Merrill

Numerade Educator

Problem 21

Daily oil imports to the United States from Mexico can be approximated by
$$
I(t)=-0.015 t^{2}+0.1 t+1.4
$$
million barrels $/$ day $(0 \leq t \leq 8)$ where $t$ is time in years since the start of $2000 .^{2}$ According to the model, in what year were oil imports to the United States greatest? How many barrels per day were imported that year? HINT [See Example 1.]

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 22

Daily oil production by Pemex, Mexico's national oil company, for 2001-2009 can be approximated by
$$
P(t)=-0.022 t^{2}+0.2 t+2.9
$$
million barrels $/$ day $(1 \leq t \leq 9)$ where $t$ is time in years since the start of $2000 .{ }^{3}$ According to the model, in what year was oil production by Pemex greatest? How many barrels per day were produced that year?

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 23

The fuel efficiency (in miles per gallon) of an SUV depends on its weight according to the formula $^{4}$
$$
E=0.0000016 x^{2}-0.016 x+54 \quad(1,800 \leq x \leq 5,400)
$$
where $x$ is the weight of an SUV in pounds. According to the model, what is the weight of the least fuel-efficient SUV? Would you trust the model for weights greater than the answer you obtained? Explain.

Chasen Shaw

Numerade Educator

Problem 24

The amount of carbon dioxide (in pounds per 15,000 cubic miles) released by a typical SUV depends on its fuel efficiency according to the formula $^{5}$
$$
W=32 x^{2}-2,080 x+44,000 \quad(12 \leq x \leq 33)
$$
where $x$ is the fuel efficiency of an SUV in miles per gallon. According to the model, what is the fuel efficiency of the SUV with the least carbon dioxide pollution? Comment on the reliability of the model for fuel efficiencies that exceed your answer.

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 25

The market research department of the Better Baby Buggy Co. predicts that the demand equation for its buggies is given by $q=-0.5 p+140$, where $q$ is the number of buggies it can sell in a month if the price is $\$ p$ per buggy. At what price should it sell the buggies to get the largest revenue? What is the largest monthly revenue?

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 26

The Better Baby Buggy Co. has just come out with a new model, the Turbo. The market research department predicts that the demand equation for Turbos is given by $q=-2 p+320$, where $q$ is the number of buggies it can sell in a month if the price is $\$ p$ per buggy. At what price should it sell the buggies to get the largest revenue? What is the largest monthly revenue?

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 27

Pack-Em-In Real Estate is building a new housing development. The more houses it builds, the less people will be willing to pay, due to the crowding and smaller lot sizes. In fact, if it builds 40 houses in this particular development, it can sell them for $$\$ 200,000$$ each, but if it builds 60 houses, it will only be able to get $$\$ 160,000$$ each. Obtain a linear demand equation and hence determine how many houses PackEm-In should build to get the largest revenue. What is the largest possible revenue?

Carson Merrill

Numerade Educator

Problem 28

Pack-Em-In has another development in the works. If it builds 50 houses in this development, it will be able to sell them at $$\$ 190,000$$ each, but if it builds 70 houses, it will get only $$\$ 170,000$$ each. Obtain a linear demand equation and hence determine how many houses it should build to get the largest revenue. What is the largest possible revenue?

Carson Merrill

Numerade Educator

Problem 29

In 2005, the Las Vegas monorail charged $$\$ 3$$ per ride and had an average ridership of about 28,000 per day. In December 2005 the Las Vegas Monorail Company raised the fare to $$\$ 5$$ per ride, and average ridership in 2006 plunged to around 19,000 per day. ${ }^{6}$
a. Use the given information to find a linear demand equation.
b. Find the price the company should have charged to maximize revenue from ridership. What is the corresponding daily revenue?
c. The Las Vegas Monorail Company would have needed $$\$ 44.9$$ million in revenues from ridership to break even in $2006 .$ Would it have been possible to break even in 2006 by charging a suitable price?

Carson Merrill

Numerade Educator

Problem 30

The Utarek monorail, which links the three urbynes (or districts) of Utarek, Mars, charged $\overline{\bar{Z}} 5$ per ride $^{7}$ and sold about 14 million rides per day. When the Utarek City Council lowered the fare to $\overline{\bar{Z}}_{3}$ per ride, the number of rides increased to 18 million per day.
a. Use the given information to find a linear demand equation.
b. Find the price the City Council should have charged to maximize revenue from ridership. What is the corresponding daily revenue?
c. The City Council would have needed to raise $\overline{\bar{Z}}_{4} 8$ billion in revenues from ridership each Martian year ( 670 days $^{8}$ ) to finance the new Mars organism research lab. Would this have been possible by charging a suitable price?

Carson Merrill

Numerade Educator

Problem 31

You operate a gaming Web site, www .mudbeast.net, where users must pay a small fee to log on. When you charged $$\$ 2$$ the demand was 280 log-ons per month. When you lowered the price to $$\$ 1.50$$, the demand increased to 560 logons per month.
a. Construct a linear demand function for your Web site and hence obtain the monthly revenue $R$ as a function of the log-on fee $x$.
b. Your Internet provider charges you a monthly fee of $$\$ 30$$ to maintain your site. Express your monthly profit $P$ as a function of the log-on fee $x$, and hence determine the logon fee you should charge to obtain the largest possible monthly profit. What is the largest possible monthly profit? HINT [See Example 4.]

Carson Merrill

Numerade Educator

Problem 32

Two fraternities, Sig Ep and Ep Sig, plan to raise money jointly to benefit homeless people on Long Island. They will sell Yoda vs. Alien T-shirts in the student center, but are not sure how much to charge. Sig Ep treasurer Augustus recalls that they once sold 400 shirts in a week at $$\$ 8$$ per shirt, but Ep Sig treasurer Julius has solid research indicating that it is possible to sell 600 per week at $$\$ 4$$ per shirt.
a. Based on this information, construct a linear demand equation for Yoda vs. Alien T-shirts, and hence obtain the weekly revenue $R$ as a function of the unit price $x$.
b. The university administration charges the fraternities a weekly fee of $$\$ 500$$ for use of the Student Center. Write down the monthly profit $P$ as a function of the unit price $x$, and hence determine how much the fraternities should charge to obtain the largest possible weekly profit. What is the largest possible weekly profit? HINT [See Example 4.]

Carson Merrill

Numerade Educator

Problem 33

The latest demand equation for your gaming Web site, www.mudbeast.net, is given by
$$
q=-400 x+1,200
$$
where $q$ is the number of users who $\log$ on per month and $x$ is the log-on fee you charge. Your Internet provider bills you as follows:
$$
\begin{array}{ll}
\text { Site maintenance fee: } & \$ 20 \text { per month } \\
\text { High-volume access fee: } & 50 \varnothing \text { per log-on }
\end{array}
$$
Find the monthly cost as a function of the log-on fee $x$. Hence, find the monthly profit as a function of $x$ and determine the log-on fee you should charge to obtain the largest possible monthly profit. What is the largest possible monthly profit?

Carson Merrill

Numerade Educator

Problem 34

The latest demand equation for your Yoda vs. Alien T-shirts is given by
$$
q=-40 x+600
$$
where $q$ is the number of shirts you can sell in one week if you charge $$\$ x$$ per shirt. The Student Council charges you $$\$ 400$$ per week for use of their facilities, and the T-shirts cost you $$\$ 5$$ each. Find the weekly cost as a function of the unit price $x$. Hence, find the weekly profit as a function of $x$ and determine the unit price you should charge to obtain the largest possible weekly profit. What is the largest possible weekly profit?

Carson Merrill

Numerade Educator

Problem 35

You have just opened a new nightclub, Russ' Techno Pitstop, but are unsure of how high to set the cover charge (entrance fee). One week you charged $$\$ 10$$ per guest and averaged 300 guests per night. The next week you charged $$\$ 15$$ per guest and averaged 250 guests per night.
a. Find a linear demand equation showing the number of guests $q$ per night as a function of the cover charge $p$.
b. Find the nightly revenue $R$ as a function of the cover charge $p$.
c. The club will provide two free non-alcoholic drinks for each guest, costing the club $$\$ 3$$ per head. In addition, the nightly overheads (rent, salaries, dancers, DJ, etc.) amount to $$\$ 3,000$$. Find the cost $C$ as a function of the cover charge $p$.
d. Now find the profit in terms of the cover charge $p$, and hence determine the entrance fee you should charge for a maximum profit.

Carson Merrill

Numerade Educator

Problem 36

As sales manager for Montevideo Productions, Inc., you are planning to review the prices you charge clients for television advertisement development. You currently charge each client an hourly development fee of $$\$ 2,500$$. With this pricing structure, the demand, measured by the number of contracts Montevideo signs per month, is 15 contracts. This is down 5 contracts from the figure last year, when your company charged only $$\$ 2,000$$.
a. Construct a linear demand equation giving the number of contracts $q$ as a function of the hourly fee $p$ Montevideo charges for development.
b. On average, Montevideo bills for 50 hours of production time on each contract. Give a formula for the total revenue obtained by charging $$\$ p$$ per hour.
c. The costs to Montevideo Productions are estimated as follows:
$$
\begin{array}{ll}
\text { Fixed costs: } & \$ 120,000 \text { per month } \\
\text { Variable costs: } & \$ 80,000 \text { per contract }
\end{array}
$$
Express Montevideo Productions' monthly cost (i) as a function of the number $q$ of contracts and (ii) as a function of the hourly production charge $p$.
d. Express Montevideo Productions' monthly profit as a function of the hourly development fee $p$ and hence the price it should charge to maximize the profit.

Carson Merrill

Numerade Educator

Problem 37

The following table shows total military and arms trade expenditure in 1994,1998 , and 2006. (See Exercise $19 ; t=4$ represents 1994.) $^{9}$
$$
\begin{array}{|r|c|c|c|}
\hline \text { Year } \boldsymbol{t} & 4 & 8 & 16 \\
\hline \text { Military Expenditure (\$ billion) } & 900 & 800 & 1,200 \\
\hline
\end{array}
$$
Find a quadratic model for these data, and use your model to estimate world military expenditure in 2008 . Compare your answer with the actual figure shown in Exercise $19 .$ HINT [See Example 5.]

Carson Merrill

Numerade Educator

Problem 38

The following table shows the percentage of the U.S. Discretionary Budget allocated to education in 2003,2005, and 2009 . (See Exercise $20 ; t=3$ represents the start of $2003 .)^{10}$
$$
\begin{array}{|r|c|c|c|}
\hline \text { Year } \boldsymbol{t} & 3 & 5 & 9 \\
\hline \text { Percentage } & 6.8 & 7 & 6.2 \\
\hline
\end{array}
$$
Find a quadratic model for these data, and use your model to estimate the percentage of the U.S. Discretionary Budget allocated to education in 2008 . Compare your answer with the actual figure shown in Exercise 20 .

Carson Merrill

Numerade Educator

Problem 39

The following table shows Apple iPhone sales from the 2nd quarter in 2007 through the second quarter in $2008(t=2$ represents the second quarter of 2007$):^{.11}$
$$
\begin{array}{|r|c|c|c|c|c|}
\hline \text { Quarter } \boldsymbol{t} & 2 & 3 & 4 & 5 & 6 \\
\hline \begin{array}{r}
\text { iPhone Sales } \\
\text { (thousands) }
\end{array} & 270 & 1,119 & 2,315 & 1,703 & 717 \\
\hline
\end{array}
$$
a. Find a quadratic regression model for these data. (Round coefficients to the nearest whole number.) Graph the model together with the data.
b. What does the model predict for iPhone sales in the third quarter of $2008(t=7)$ to the nearest 1,000 units? Comment on the answer, and ascertain the actual third quarter sales in 2008 ( Apple's fiscal fourth quarter).

Carson Merrill

Numerade Educator

Problem 40

The following table gives the approximate number of Facebook users at various times since its establishment early in $2004 .^{12}$
$$
\begin{array}{|r|r|r|r|r|r|r|r|r|}
\hline \begin{array}{r}
\text { Year } \boldsymbol{t} \\
\text { (since start } \\
\text { of 2004) }
\end{array} & 0 & 1 & 2 & 2.5 & 3 & 3.5 & 4 & 4.5 \\
\hline \begin{array}{r}
\text { Facebook } \\
\text { Members } \boldsymbol{n} \\
\text { (millions) }
\end{array} & 0 & 1 & 5.5 & 7 & 12 & 30 & 58 & 80 \\
\hline
\end{array}
$$
a. Find a quadratic regression model for these data. (Round coefficients to the nearest whole number.) Graph the model, together with the data.
b. Assuming the trend had continued, estimate the number of members at the start of 2010 to the nearest 10 million members.
c. Is the quadratic model appropriate for long-term prediction of the number of members? Why?

Carson Merrill

Numerade Educator

Problem 41

What can you say about the graph of $f(x)=a x^{2}+b x+c$ if $a=0 ?$

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 42

What can you say about the graph of $f(x)=a x^{2}+b x+c$ if $c=0$ ?

Charles Machakwa

Charles Machakwa

Numerade Educator

Problem 43

Following is the graph of $f(x)=a x^{2}+$ $b x+c:$
(A) $a$ is positive and $c$ is positive.
(B) $a$ is negative and $c$ is positive.
(C) $a$ is positive and $c$ is negative.
(D) $a$ is negative and $c$ is negative.

Carson Merrill

Numerade Educator

Problem 44

Following is the graph of $f(x)=$ $a x^{2}+b x+c:$
(A) $a$ is positive and $c$ is positive.
(B) $a$ is negative and $c$ is positive.
(C) $a$ is positive and $c$ is negative.
(D) $a$ is negative and $c$ is negative.

Carson Merrill

Numerade Educator

Problem 45

Refer to the graph of $f(x)=a x^{2}+b x+c$ in Exercise 43 . Is $b$ positive or negative? Why?

Carson Merrill

Numerade Educator

Problem 46

Refer to the graph of $f(x)=a x^{2}+b x+c$ in Exercise 44 . Is $b$ positive or negative? Why?

Carson Merrill

Numerade Educator

Problem 47

Suppose the graph of revenue as a function of unit price is a parabola that is concave down. What is the significance of the coordinates of the vertex, the $x$ -intercepts, and the $y$ -intercept?

Carson Merrill

Numerade Educator

Problem 48

Suppose the height of a stone thrown vertically upward is given by a quadratic function of time. What is the significance of the coordinates of the vertex, the (possible) $x$ -intercepts, and the $y$ -intercept?

Carson Merrill

Numerade Educator

Problem 49

How might you tell, roughly, whether a set of data should be modeled by a quadratic rather than by a linear equation?

Carson Merrill

Numerade Educator

Problem 50

A member of your study group tells you that, because the following set of data does not suggest a straight line, the data are best modeled by a quadratic.
$$
\begin{array}{|c|c|c|c|c|c|}
\hline \boldsymbol{x} & 0 & 2 & 4 & 6 & 8 \\
\hline \boldsymbol{y} & 1 & 2 & 1 & 0 & 1 \\
\hline
\end{array}
$$
Comment on her suggestion.

Carson Merrill

Numerade Educator

Problem 51

Is a quadratic model useful for long-term prediction of sales of an item? Why?

Carson Merrill

Numerade Educator

Problem 52

Of what use is a quadratic model, if not for long-term prediction?

Carson Merrill

Numerade Educator

Problem 53

Explain why, if demand is a linear function of unit price $p$ (with negative slope), then there must be a single value of $p$ that results in the maximum revenue.

Carson Merrill

Numerade Educator

Problem 54

Explain why, if the average cost of a commodity is given by $y=0.1 x^{2}-4 x-2$, where $x$ is the number of units sold, there is a single choice of $x$ that results in the lowest possible average cost.

Carson Merrill

Numerade Educator

Problem 55

If the revenue function for a particular commodity is $R(p)=-50 p^{2}+60 p$, what is the (linear) demand function? Give a reason for your answer.

Carson Merrill

Numerade Educator

Problem 56

If the revenue function for a particular commodity is $R(p)=-50 p^{2}+60 p+50$, can the demand function be linear? What is the associated demand function?

Carson Merrill

Numerade Educator