Chapter 1

The number of passengers using a railway fell from 190,205 to 174,989 during a 5 -year period. Find the annual percentage decrease over this period.

Table \(1.15\) shows world population, \(P\), in billions of people, world passenger automobile production, \(A\), in millions of cars, and world cell phone subscribers, \(C\), in millions of subscribers. \({ }^{33}\) (a) Find the average rate of change, with units, for each of \(P, A\), and \(C\) between 1995 and 2005 . (b) Between 1995 and 2005, which increased faster: (i) Population or the number of automobiles? (ii) Population or the number of cell phone subscribers? $$ \begin{array}{l} \text { Table } 1.15\\\ \begin{array}{c|c|c|c} \hline \text { Year } & 1995 & 2000 & 2005 \\ \hline P \text { (billions) } & 5.68 & 6.07 & 6.45 \\ \hline A \text { (millions) } & 36.1 & 41.3 & 45.9 \\ \hline C \text { (millions) } & 91 & 740 & 2168 \\ \hline \end{array} \end{array} $$

The following formulas give the populations of four different towns, \(A, B, C\), and \(D\), with \(t\) in years from now. $$ \begin{array}{cl} P_{A}=600 e^{0.09 t} & P_{B}=1000 e^{-0.02 t} \\ P_{C}=1200 e^{0.03 t} & P_{D}=900 e^{0.12 t} \end{array} $$ (a) Which town is growing fastest (that is, has the largest percentage growth rate)? (b) Which town is the largest now? (c) Are any of the towns decreasing in size? If so, which one(s)?

A $$\$ 50,000$$ tractor has a resale value of $$\$ 10,000$$ twenty years after it was purchased. Assume that the value of the tractor depreciates linearly from the time of purchase. (a) Find a formula for the value of the tractor as a function of the time since it was purchased. (b) Graph the value of the tractor against time. (c) Find the horizontal and vertical intercepts, give units, and interpret them.

The company that produces Cliffs Notes (abridged versions of classic literature) was started in 1958 with $$\$ 4000$$ and sold in 1998 for $$\$ 14,000,000$$. Find the annual percent increase in the value of this company over the 40 years.

Financial investors know that, in general, the higher the expected rate of return on an investment, the higher the corresponding risk. (a) Graph this relationship, showing expected return as a function of risk. (b) On the figure from part (a), mark a point with high expected return and low risk. (Investors hope to find such opportunities.)

The percentage of people, \(P\), below the poverty level in the \(\mathrm{US}^{14}\) is given in Table \(1.4 .\) (a) Find a formula for the percentage in poverty as a linear function of time in years since 2000 . (b) Use the formula to predict the percentage in poverty in 2006 . (c) What is the difference between the prediction and the actual percentage, \(12.3 \%\) ? $$ \begin{array}{l|c|c|c|c} \hline \text { Year (since 2000) } & 0 & 1 & 2 & 3 \\ \hline P \text { (percentage) } & 11.3 & 11.7 & 12.1 & 12.5 \\ \hline \end{array} $$

The blood mass of a mammal is proportional to its body mass. A rhinoceros with body mass 3000 kilograms has blood mass of 150 kilograms. Find a formula for the blood mass of a mammal as a function of the body mass and estimate the blood mass of a human with body mass 70 kilograms.

An exponentially growing animal population numbers 500 at time \(t=0\); two years later, it is \(1500 .\) Find a formula for the size of the population in \(t\) years and find the size of the population at \(t=5\).

A city's population is 1000 and growing at \(5 \%\) a year. (a) Find a formula for the population at time \(t\) years from now assuming that the \(5 \%\) per year is an: (i) Annual rate (ii) Continuous annual rate (b) In each case in part (a), estimate the population of the city in 10 years.

Find a formula for the number of zebra mussels in a bay as a function of the number of years since 2003 , given that there were 2700 at the start of 2003 and 3186 at the start of 2004 . (a) Assume that the number of zebra mussels is growing linearly. Give units for the slope of the line and interpret it in terms of zebra mussels. (b) Assume that the number of zebra mussels is growing exponentially. What is the percent rate of growth of the zebra mussel population?

World grain production was 1241 million tons in 1975 and 2048 million tons in 2005, and has been increasing at an approximately constant rate. \({ }^{15}\) (a) Find a linear function for world grain production, \(P\), in million tons, as a function of \(t\), the number of years since 1975 . (b) Using units, interpret the slope in terms of grain production. (c) Using units, interpret the vertical intercept in terms of grain production. (d) According to the linear model, what is the predicted world grain production in 2015 ? (e) According to the linear model, when is grain production predicted to reach 2500 million tons?

Biologists estimate that the number of animal species of a certain body length is inversely proportional to the square of the body length. \({ }^{69}\) Write a formula for the number of animal species, \(N\), of a certain body length as a function of the length, \(L\). Are there more species at large lengths or at small lengths? Explain.

A tree of height \(y\) meters has, on average, \(B\) branches, where \(B=y-1 .\) Each branch has, on average, \(n\) leaves where \(n=2 B^{2}-B .\) Find the average number of leaves of a tree as a function of height.

In the form \(P=P_{0} a^{t}\). Which represent exponential growth and which represent exponential decay? $$P=15 e^{0.25 t}$$

Worldwide, wind energy \(^{53}\) generating capacity, \(W\), was 40,300 megawatts in 2003 and 121,188 megawatts in 2008 . (a) Use the values given to write \(W\), in megawatts, as a linear function of \(t\), the number of years since 2003 . (b) Use the values given to write \(W\) as an exponential function of \(t\) (c) Graph the functions found in parts (a) and (b) on the same axes. Label the given values.

At time \(t\) in seconds, a particle's distance \(s(t)\), in \(\mathrm{cm}\), from a point is given in the table. What is the average velocity of the particle from \(t=3\) to \(t=10 ?\) $$ \begin{array}{c|c|c|c|c|c} \hline t & 0 & 3 & 6 & 10 & 13 \\ \hline s(t) & 0 & 72 & 92 & 144 & 180 \\ \hline \end{array} $$

Search and rescue teams work to find lost hikers. Members of the search team separate and walk parallel to one another through the area to be searched. Table \(1.5\) shows the percent, \(P\), of lost individuals found for various separation distances, \(d\), of the searchers. 16 $$ \begin{array}{l|lllll} \hline \text { Separation distance } d \text { (ft) } & 20 & 40 & 60 & 80 & 100 \\ \hline \text { Approximate percent found, } P & 90 & 80 & 70 & 60 & 50 \\ \hline \end{array} $$ (a) Explain how you know that the percent found, \(P\), could be a linear function of separation distance, \(d\). (b) Find \(P\) as a linear function of \(d\). (c) What is the slope of the function? Give units and interpret the answer. (d) What are the vertical and horizontal intercepts of the function? Give units and interpret the answers.

Pregnant women metabolize some drugs at a slower rate than the rest of the population. The half-life of caffeine is about 4 hours for most people. In pregnant women, it is 10 hours. \({ }^{61}\) (This is important because caffeine, like all psychoactive drugs, crosses the placenta to the fetus.) If a pregnant woman and her husband each have a cup of coffee containing \(100 \mathrm{mg}\) of caffeine at \(8 \mathrm{am}\), how much caffeine does each have left in the body at \(10 \mathrm{pm}\) ?

In the form \(P=P_{0} a^{t}\). Which represent exponential growth and which represent exponential decay? $$P=2 e^{-0.5 t}$$

(a) Which (if any) of the functions in the following table could be linear? Find formulas for those functions. (b) Which (if any) of these functions could be exponential? Find formulas for those functions. $$ \begin{array}{r|c|c|c} \hline \multicolumn{1}{r|} {x} & f(x) & g(x) & h(x) \\ \hline-2 & 12 & 16 & 37 \\ -1 & 17 & 24 & 34 \\ 0 & 20 & 36 & 31 \\ 1 & 21 & 54 & 28 \\ 2 & 18 & 81 & 25 \\ \hline \end{array} $$

Table \(1.16\) shows the production of tobacco in the US. 34 (a) What is the average rate of change in tobacco production between 1996 and \(2003 ?\) Give units and interpret your answer in terms of tobacco production. (b) During this seven-year period, is there any interval during which the average rate of change was positive? If so, when? $$ \begin{array}{l} \text { Table 1.16 Tobacco production, in millions of pounds }\\\ \begin{array}{c|c|c|c|c|c|c|c|c} \hline \text { Year } & 1996 & 1997 & 1998 & 1999 & 2000 & 2001 & 2002 & 2003 \\\ \hline \text { Production } & 1517 & 1787 & 1480 & 1293 & 1053 & 991 & 879 & 831 \\ \hline \end{array} \end{array} $$

Annual sales of music compact discs (CDs) have declined since 2000 . Sales were \(942.5\) million in 2000 and \(384.7\) million in \(2008 .^{17}\) (a) Find a formula for annual sales, \(S\), in millions of music CDs, as a linear function of the number of years, \(t\), since 2000 . (b) Give units for and interpret the slope and the vertical intercept of this function. (c) Use the formula to predict music \(\mathrm{CD}\) sales in 2012 .

The circulation time of a mammal (that is, the average time it takes for all the blood in the body to circulate once and return to the heart) is proportional to the fourth root of the body mass of the mammal. (a) Write a formula for the circulation time, \(T\), in terms of the body mass, \(B\). (b) If an elephant of body mass 5230 kilograms has a circulation time of 148 seconds, find the constant of proportionality. (c) What is the circulation time of a human with body mass 70 kilograms?

The half-life of radioactive strontium-90 is 29 years. In 1960, radioactive strontium-90 was released into the atmosphere during testing of nuclear weapons, and was absorbed into people's bones. How many years does it take until only \(10 \%\) of the original amount absorbed remains?

In the form \(P=P_{0} a^{t}\). Which represent exponential growth and which represent exponential decay? $$P=P_{0} e^{0.2 t}$$

A company produces and sells shirts. The fixed costs are $$\$ 7000$$ and the variable costs are $$\$ 5$$ per shirt. (a) Shirts are sold for $$\$ 12$$ each. Find cost and revenue as functions of the quantity of shirts, \(q\). (b) The company is considering changing the selling price of the shirts. Demand is \(q=2000-40 p\), where \(p\) is price in dollars and \(q\) is the number of shirts. What quantity is sold at the current price of $$\$ 12 ?$$ What profit is realized at this price? (c) Use the demand equation to write cost and revenue as functions of the price, \(p\). Then write profit as a function of price. (d) Graph profit against price. Find the price that maximizes profits. What is this profit?

Determine whether each of the following tables of values could correspond to a linear function, an exponential function, or neither. For each table of values that could correspond to a linear or an exponential function, find a formula for the function. $$ \begin{array}{l} \text { (a) }\\\ \begin{array}{l|l} \hline x & f(x) \\ \hline 0 & 10.5 \\ 1 & 12.7 \\ 2 & 18.9 \\ 3 & 36.7 \\ \hline \end{array} \end{array} $$ $$ \begin{array}{l} \text { (b) }\\\ \begin{array}{c|l} \hline t & s(t) \\ \hline-1 & 50.2 \\ 0 & 30.12 \\ 1 & 18.072 \\ 2 & 10.8432 \\ \hline \end{array} \end{array} $$ $$ \begin{array}{l} \text { (c) }\\\ \begin{array}{c|c} \hline u & g(u) \\ \hline 0 & 27 \\ 2 & 24 \\ 4 & 21 \\ 6 & 18 \\ \hline \end{array} \end{array} $$

Do you expect the average rate of change (in units per year) of each of the following to be positive or negative? Explain your reasoning. (a) Number of acres of rain forest in the world. (b) Population of the world. (c) Number of polio cases each year in the US, since \(1950 .\) (d) Height of a sand dune that is being eroded. (e) Cost of living in the US.

In a California town, the monthly charge for waste collection is \(\$ 8\) for 32 gallons of waste and \(\$ 12.32\) for 68 gallons of waste. (a) Find a linear formula for the cost, \(C\), of waste collection as a function of the number of gallons of waste, \(w .\) (b) What is the slope of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection. (c) What is the vertical intercept of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection.

The DuBois formula relates a person's surface area \(s\), in \(\mathrm{m}^{2}\), to weight \(w\), in \(\mathrm{kg}\), and height \(h\), in \(\mathrm{cm}\), by $$ s=0.01 w^{0.25} h^{0.75} $$ (a) What is the surface area of a person who weighs \(65 \mathrm{~kg}\) and is \(160 \mathrm{~cm}\) tall? (b) What is the weight of a person whose height is \(180 \mathrm{~cm}\) and who has a surface area of \(1.5 \mathrm{~m}^{2}\) ? (c) For people of fixed weight \(70 \mathrm{~kg}\), solve for \(h\) as a function of \(s\). Simplify your answer.

In 1923, koalas were introduced on Kangaroo Island off the coast of Australia. In 1996 , the population was 5000 . By 2005, the population had grown to 27,000 , prompting a debate on how to control their growth and avoid koalas dying of starvation. \(^{62}\) Assuming exponential growth, find the (continuous) rate of growth of the koala population between 1996 and \(2005 .\) Find a formula for the population as a function of the number of years since 1996, and estimate the population in the year 2020 .

In the form \(P=P_{0} a^{t}\). Which represent exponential growth and which represent exponential decay? $$P=7 e^{-\pi t}$$

When the price, \(p\), charged for a boat tour was $$\$ 25$$, the average number of passengers per week, \(N\), was 500 . When the price was reduced to $$\$ 20$$, the average number of passengers per week increased to \(650 .\) Find a formula for the demand curve, assuming that it is linear.

(a) Could the data on annual world soybean production \(^{54}\) in Table \(1.32\) correspond to a linear function or an exponential function? If so, which? (b) Find a formula for \(P\), world soybean production in millions of tons, as a function of time, \(t\), in years since 2000 . (c) What is the annual percent increase in soybean production? $$ \begin{array}{l} \text { Table 1.32 Soybean production, in millions of tons }\\\ \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 \\ \hline \text { Production } & 161.0 & 170.3 & 180.2 & 190.7 & 201.8 & 213.5 \\\ \hline \end{array} \end{array} $$

The number of species of coastal dune plants in Australia decreases as the latitude, in \({ }^{\circ} \mathrm{S}\), increases. There are 34 species at \(11^{\circ} \mathrm{S}\) and 26 species at \(44^{\circ} \mathrm{S} .^{18}\) (a) Find a formula for the number, \(N\), of species of coastal dune plants in Australia as a linear function of the latitude, \(l\), in \({ }^{\circ} \mathrm{S}\). (b) Give units for and interpret the slope and the vertical intercept of this function. (c) Graph this function between \(l=11^{\circ} \mathrm{S}\) and \(l=\) \(44^{\circ} \mathrm{S}\). (Australia lies entirely within these latitudes.)

The infrastructure needs of a region (for example, the number of miles of electrical cable, the number of miles of roads, the number of gas stations) depend on its population. Cities enjoy economies of scale. \({ }^{70}\) For example, the number of gas stations is proportional to the population raised to the power of \(0.77\). (a) Write a formula for the number, \(N\), of gas stations in a city as a function of the population, \(P\), of the city. (b) If city \(A\) is 10 times bigger than city \(B\), how do their number of gas stations compare? (c) Which is expected to have more gas stations per person, a town of 10,000 people or a city of 500,000 people?

The total world marine catch in 1950 was 17 million tons and in 2001 was 99 million tons. \(^{63}\) If the marine catch is increasing exponentially, find the (continuous) rate of increase. Use it to predict the total world marine catch in the year 2020 .

Put the functions in the form \(P=P_{0} e^{k t}\). $$P=15(1.5)^{t}$$

Table \(1.28\) gives data for the linear demand curve for a product, where \(p\) is the price of the product and \(q\) is the quantity sold every month at that price. Find formulas for the following functions. Interpret their slopes in terms of demand. (a) \(q\) as a function of \(p\). (b) \(p\) as a function of \(q\). $$ \begin{array}{l} \text { Table } 1.28\\\ \begin{array}{c|c|c|c|c|c} \hline p \text { (dollars) } & 16 & 18 & 20 & 22 & 24 \\ \hline q \text { (tons) } & 500 & 460 & 420 & 380 & 340 \\ \hline \end{array} \end{array} $$

The 2004 US presidential debates questioned whether the minimum wage has kept pace with inflation. Decide the question using the following information: \(^{.55}\) In 1938 , the minimum wage was \(25 d\); in 2004 , it was $$\$ 5.15$$. During the same period, inflation averaged \(4.3 \%\).

Table \(1.17\) shows the total US labor force, \(L\). Find the average rate of change between 1940 and 2000 ; between 1940 and 1960 ; between 1980 and 2000 . Give units and interpret your answers in terms of the labor force. \({ }^{36}\) $$ \begin{array}{l} \text { Table 1.17 US labor force, in thousands of workers }\\\ \begin{array}{c|c|c|c|c} \hline \text { Year } & 1940 & 1960 & 1980 & 2000 \\ \hline L & 47,520 & 65,778 & 99,303 & 136,891 \\ \hline \end{array} \end{array} $$

Table \(1.6\) gives the average weight, \(w\), in pounds, of American men in their sixties for height, \(h\), in inches. (a) How do you know that the data in this table could represent a linear function? (b) Find weight, \(w\), as a linear function of height, \(h\). What is the slope of the line? What are the units for the slope? (c) Find height, \(h\), as a linear function of weight, \(w\). What is the slope of the line? What are the units for the slope? $$ \begin{array}{l|c|c|c|c|c|c|c|c} \hline h \text { (inches) } & 68 & 69 & 70 & 71 & 72 & 73 & 74 & 75 \\ \hline w \text { (pounds) } & 166 & 171 & 176 & 181 & 186 & 191 & 196 & 201 \\\ \hline \end{array} $$

(a) Use the Rule of 70 to predict the doubling time of an investment which is earning \(8 \%\) interest per year. (b) Find the doubling time exactly, and compare your answer to part (a).

Put the functions in the form \(P=P_{0} e^{k t}\). $$P=10(1.7)^{t}$$

The demand curve for a product is given by \(q=\) \(120,000-500 p\) and the supply curve is given by \(q=\) \(1000 p\) for \(0 \leq q \leq 120,000\), where price is in dollars. (a) At a price of $$\$ 100$$, what quantity are consumers willing to buy and what quantity are producers willing to supply? Will the market push prices up or down? (b) Find the equilibrium price and quantity. Does your answer to part (a) support the observation that market forces tend to push prices closer to the equilibrium price?

(a) Niki invested $$\$ 10,000$$ in the stock market. The investment was a loser, declining in value \(10 \%\) per year each year for 10 years. How much was the investment worth after 10 years? (b) After 10 years, the stock began to gain value at \(10 \%\) per year. After how long will the investment regain its initial value $$(\$ 10,000)$$ ?

The total world marine catch \(^{37}\) of fish, in metric tons, was 17 million in 1950 and 99 million in 2001 . What was the average rate of change in the marine catch during this period? Give units and interpret your answer.

Concern the maximum heart rate (MHR), which is the maximum number of times a person's heart can safely beat in one minute. If MHR is in beats per minute and \(a\) is age in years, the formulas used to estimate MHR, are $$ \begin{array}{c} \text { For females: } \mathrm{MHR}=226-a, \\ \text { For males: } \mathrm{MHR}=220-a \end{array} $$ Which of the following is the correct statement? (a) As you age, your maximum heart rate decreases by one beat per year. (b) As you age, your maximum heart rate decreases by one beat per minute. (c) As you age, your maximum heart rate decreases by one beat per minute per year.

A standard tone of 20,000 dynes/cm \(^{2}\) (about the loudness of a rock band) is assigned a value of 10 . A subject listened to other sounds, such as a light whisper, normal conversation, thunder, a jet plane at takeoff, and so on. In each case, the subject was asked to judge the loudness and assign it a number relative to 10 , the value of the standard tone. This is a "judgment of magnitude" experiment. The power law \(J=a l^{0.3}\) was found to model the situation well, where \(l\) is the actual loudness (measured in dynes/cm \(^{2}\) ) and \(J\) is the judged loudness. (a) What is the value of \(a\) ? (b) What is the judged loudness if the actual loudness is \(0.2\) dynes \(/ \mathrm{cm}^{2}\) (normal conversation)? (c) What is the actual loudness if judged loudness is \(20 ?\)