Chapter 1

The island of Manhattan was sold for $$\$ 24$$ in \(1626 .\) Suppose the money had been invested in an account which compounded interest continuously. (a) How much money would be in the account in the year 2005 if the yearly interest rate was (i) \(5 \% ?\) (ii) \(7 \%\) ? (b) If the yearly interest rate was \(6 \%\), in what year would the account be worth one million dollars?

Put the functions in the form \(P=P_{0} e^{k t}\). $$P=174(0.9)^{t}$$

World production, \(Q\), of zinc in thousands of metric tons and the value, \(P\), in dollars per metric ton are given 8 in Table \(1.29\). Plot the value as a function of production. Sketch a possible supply curve. $$ \begin{array}{l} \text { Table 1.29 } \quad \text { World zinc production }\\\ \begin{array}{c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 \\ \hline Q & 9520 & 9590 & 9930 & 10,000 & 10,900 \\ \hline P & 896 & 1160 & 1480 & 3500 & 3400 \\ \hline \end{array} \end{array} $$

A photocopy machine can reduce copies to \(80 \%\) of their original size. By copying an already reduced copy, further reductions can be made. (a) If a page is reduced to \(80 \%\), what percent enlargement is needed to return it to its original size? (b) Estimate the number of times in succession that a page must be copied to make the final copy less than \(15 \%\) of the size of the original.

Table \(1.18\) gives the revenues, \(R\), of General Motors, formerly the world's largest auto manufacturer. \({ }^{38}\) (a) Find the change in revenues between 2003 and 2008 . (b) Find the average rate of change in revenues between 2003 and 2008. Give units and interpret your answer. (c) From 2003 to 2008 , were there any one-year intervals during which the average rate of change was negative? If so, which? $$ \begin{array}{l} \text { Table 1.18 GM revenues, billions of dollars }\\\ \begin{array}{l|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 \\ \hline R & 184.0 & 192.9 & 193.1 & 205.6 & 181.1 & 149.0 \\ \hline \end{array} \end{array} $$

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 for a male and female of the same age? (a) Their maximum heart rates are the same. (b) The male's maximum heart rate exceeds the female's. (c) The female's maximum heart rate exceeds the male's.

The desert temperature, \(H\), oscillates daily between \(40^{\circ} \mathrm{F}\) at \(5 \mathrm{am}\) and \(80^{\circ} \mathrm{F}\) at \(5 \mathrm{pm}\). Write a possible formula for \(H\) in terms of \(t\), measured in hours from 5 am.

A sporting goods wholesaler finds that when the price of a product is \(\$ 25\), the company sells 500 units per week. When the price is \(\$ 30\), the number sold per week decreases to 460 units. (a) Find the demand, \(q\), as a function of price, \(p\), assuming that the demand curve is linear. (b) Use your answer to part (a) to write revenue as a function of price. (c) Graph the revenue function in part (b). Find the price that maximizes revenue. What is the revenue at this price?

In 2004 , the world's population was \(6.4\) billion, and the population was projected to reach \(8.5\) billion by the year 2030 . What annual growth rate is projected?

Put the functions in the form \(P=P_{0} e^{k t}\). $$P=4(0.55)^{t}$$

A taxi company has an annual budget of $$\$ 720,000$$ to spend on drivers and car replacement. Drivers cost the company $$\$ 30,000$$ each and car replacements cost $$\$ 20,000$$ cach. (a) What is the company's budget constraint equation? Let \(d\) be the number of drivers paid and \(c\) be the number of cars replaced. (b) Find and interpret both intercepts of the graph of the equation.

Whooping cough was thought to have been almost wiped out by vaccinations. It is now known that the vaccination wears off, leading to an increase in the number of cases, \(w\), from 1248 in 1981 to 18,957 in 2004 . (a) With \(t\) in years since 1980 , find an exponential function that fits this data. (b) What does your answer to part (a) give as the average annual percent growth rate of the number of cases? (c) On May 4, 2005, the Arizona Daily Star reported (correctly) that the number of cases had more than doubled between 2000 and 2004 . Does your model confirm this report? Explain.

The number of US households with cable television \(^{39}\) was \(12,168,450\) in 1977 and \(73,365,880\) in \(2003 .\) Estimate the average rate of change in the number of US households with cable television during this 26-year 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 $$ \begin{array}{c} \text { For females: } \mathrm{MHR}=226-a, \\ \text { For males: } \mathrm{MHR}=220-a \end{array} $$ What can be said about the ages of a male and a female with the same maximum heart rate?

A health club has cost and revenue functions given by \(C=10,000+35 q\) and \(R=p q\), where \(q\) is the number of annual club members and \(p\) is the price of a oneyear membership. The demand function for the club is \(q=3000-20 p\) (a) Use the demand function to write cost and revenue as functions of \(p\). (b) Graph cost and revenue as a function of \(p\), on the same axes. (Note that price does not go above $$\$ 170$$ and that the annual costs of running the club reach \(\$ 120,000 .)\) (c) Explain why the graph of the revenue function has the shape it does. (d) For what prices does the club make a profit? (e) Estimate the annual membership fee that maximizes profit. Mark this point on your graph.

A picture supposedly painted by Vermeer \((1632-1675)\) contains \(99.5 \%\) of its carbon-14 (half-life 5730 years). From this information decide whether the picture is a fake. Explain your reasoning.

The population of the world can be represented by \(P=\) \(6.4(1.0126)^{t}\), where \(P\) is in billions of people and \(t\) is years since 2004 . Find a formula for the population of the world using a continuous growth rate.

You have a budget of $$\$ 1000$$ for the year to cover your books and social outings. Books cost (on average) $$\$ 40$$ each and social outings cost (on average) $$\$ 10$$ each. Let \(b\) denote the number of books purchased per year and \(s\) denote the number of social outings in a year. (a) What is the equation of your budget constraint? (b) Graph the budget constraint. (It does not matter which variable you put on which axis.) (c) Find the vertical and horizontal intercepts, and give a financial interpretation for each.

Aircraft require longer takeoff distances, called takeoff rolls, at high altitude airports because of diminished air density. The table shows how the takeoff roll for a certain light airplane depends on the airport elevation. (Takeoff rolls are also strongly influenced by air temperature; the data shown assume a temperature of \(0^{\circ} \mathrm{C}\).) Determine a formula for this particular aircraft that gives the takeoff roll as an exponential function of airport elevation. $$ \begin{array}{l|c|c|c|c|c} \hline \text { Elevation (ft) } & \text { Sea level } & 1000 & 2000 & 3000 & 4000 \\ \hline \text { Takeoff roll (ft) } & 670 & 734 & 805 & 882 & 967 \\ \hline \end{array} $$

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 $$ \begin{array}{c} \text { For females: } \mathrm{MHR}=226-a, \\ \text { For males: } \mathrm{MHR}=220-a \end{array} $$ Recently \(^{20}\) it has been suggested that a more accurate predictor of MHR for both males and females is given by $$ \mathrm{MHR}=208-0.7 a $$ (a) At what age do the old and new formulas give the same MHR for females? For males? (b) Which of the following is true? (i) The new formula predicts a higher MHR for young people and a lower MHR for older people than the old formula. (ii) The new formula predicts a lower MHR for young people and a higher MHR for older people than the old formula. (c) When testing for heart disease, doctors ask patients to walk on a treadmill while the speed and incline are gradually increased until their heart rates reach 85 percent of the MHR. For a 65 -year-old male, what is the difference in beats per minute between the heart rate reached if the old formula is used and the heart rate reached if the new formula is used?

The Bay of Fundy in Canada has the largest tides in the world. The difference between low and high water levels is 15 meters (nearly 50 feet). At a particular point the depth of the water, \(y\) meters, is given as a function of time, \(t\), in hours since midnight by $$ y=D+A \cos (B(t-C)) $$ (a) What is the physical meaning of \(D\) ? (b) What is the value of \(A\) ? (c) What is the value of \(B\) ? Assume the time between successive high tides is \(12.4\) hours. (d) What is the physical meaning of \(C\) ?

Find the future value in 15 years of a $$\$ 20,000$$ payment today, if the interest rate is \(3.8 \%\) per year compounded continuously.

A fishery stocks a pond with 1000 young trout. The number of trout \(t\) years later is given by \(P(t)=1000 e^{-0.5 t}\). (a) How many trout are left after six months? After 1 year? (b) Find \(P(3)\) and interpret it in terms of trout. (c) At what time are there 100 trout left? (d) Graph the number of trout against time, and describe how the population is changing. What might be causing this?

A company has a total budget of $$\$ 500,000$$ and spends this budget on raw materials and personnel. The company uses \(m\) units of raw materials, at a cost of $$\$ 100$$ per unit, and hires \(r\) employees, at a cost of $$\$ 25,000$$ each. (a) What is the equation of the company's budget constraint? (b) Solve for \(m\) as a function of \(r\). (c) Solve for \(r\) as a function of \(m\).

Table \(1.19\) shows the concentration, \(c\), of creatinine in the bloodstream of a dog. \(^{40}\) (a) Including units, find the average rate at which the concentration is changing between the (i) \(6^{\text {th }}\) and \(8^{\text {th }}\) minutes. (ii) \(8^{\text {th }}\) and \(10^{\text {th }}\) minutes. (b) Explain the sign and relative magnitudes of your results in terms of creatinine.$$ \begin{array}{l} \text { Table } 1.19\\\ \begin{array}{c|ccccc} \hline t \text { (minutes) } & 2 & 4 & 6 & 8 & 10 \\ \hline c(\mathrm{mg} / \mathrm{ml}) & 0.439 & 0.383 & 0.336 & 0.298 & 0.266 \\\ \hline \end{array} \end{array} $$

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 $$ \begin{array}{c} \text { For females: } \mathrm{MHR}=226-a, \\ \text { For males: } \mathrm{MHR}=220-a \end{array} $$ Experiments \({ }^{21}\) suggest that the female MHR decreases by 12 beats per minute by age 21 , and by 19 beats per minute by age 33 . Is this consistent with MHR being approximately linear with age?

Find the future value in 15 years of a $$\$ 20,000$$ payment today, if the interest rate is \(3.8 \%\) per year compounded continuously.

During a recession a firm's revenue declines continuously so that the revenue, \(R\) (measured in millions of dollars), in \(t\) years' time is given by \(R=5 e^{-0.15 t}\). (a) Calculate the current revenue and the revenue in two years' time. (b) After how many years will the revenue decline to \(\$ 2.7\) million?

The population of the world reached 1 billion in 1804 , 2 billion in 1927 , 3 billion in 1960 , 4 billion in 1974 , 5 billion in 1987 and 6 billion in \(1999 .\) Find the average rate of change of the population of the world, in people per minute, during each of these intervals (that is, from 1804 to \(1927.1927\) to 1960 , etc.)

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 $$ \begin{array}{c} \text { For females: } \mathrm{MHR}=226-a, \\ \text { For males: } \mathrm{MHR}=220-a \end{array} $$ Experiments \(^{22}\) suggest that the male MHR decreases by 9 beats per minute by age 21 , and by 26 beats per minute by age \(33 .\) Is this consistent with MHR being approximately linear with age?

Find the present value of an $$\$ 8000$$ payment to be made in 5 years. The interest rate is \(4 \%\) per year compounded continuously.

(a) What is the continuous percent growth rate for \(P=\) \(100 e^{0.06 t}\), with time, \(t\), in years? (b) Write this function in the form \(P=P_{0} a^{t} .\) What is the annual percent growth rate?

A demand curve has equation \(q=100-5 p\), where \(p\) is price in dollars. A \(\$ 2\) tax is imposed on consumers. Find the equation of the new demand curve. Sketch both curves.

An Australian \(^{23}\) study found that, if other factors are constant (education, experience, etc.), taller people receive higher wages for the same work. The study reported a "height premium" for men of \(3 \%\) of the hourly wage for a \(10 \mathrm{~cm}\) increase in height; for women the height premium reported was \(2 \%\). We assume that hourly wages are a linear function of height, with slope given by the height premium at the average hourly wage for that gender. (a) The average hourly wage \(^{24}\) for a \(178 \mathrm{~cm}\) Australian man is AU$$\$29.40$$. Express the average hourly wage of an Australian man as a function of his height, \(x\) \(\mathrm{cm} .\) (b) The average hourly wage for a \(164 \mathrm{~cm}\) Australian woman is AU$$\$24.78$$. Express the average hourly wage of an Australian woman as a function of her height, \(y \mathrm{~cm}\). (c) What is the difference in average hourly wages between men and women of height \(178 \mathrm{~cm}\) ? (d) Is there a height for which men and women are predicted to have the same wage? If so, what is it?

Find the present value of a $$\$ 20,000$$ payment to be made in 10 years. Assume an interest rate of \(3.2 \%\) per year compounded continuously.

(a) What is the annual percent decay rate for \(P=\) \(25(0.88)^{t}\), with time, \(t\), in years? (b) Write this function in the form \(P=P_{0} e^{k t} .\) What is the continuous percent decay rate?

A supply curve has equation \(q=4 p-20\), where \(p\) is price in dollars. A $$\$ 2$$ tax is imposed on suppliers. Find the equation of the new supply curve. Sketch both curves.

Interest is compounded annually. Consider the following choices of payments to you: Choice 1: $$\$ 1500$$ now and $$\$ 3000$$ one year from now Choice 2: $$\$ 1900\( now and $$\$ 2500\) one year from now (a) If the interest rate on savings were \(5 \%\) per year, which would you prefer? (b) Is there an interest rate that would lead you to make a different choice? Explain.

The gross world product is \(W=32.4(1.036)^{t}\), where \(W\) is in trillions of dollars and \(t\) is years since \(2001 .\) Find a formula for gross world product using a continuous growth rate.

A tax of $$\$ 8$$ per unit is imposed on the supplier of an item. The original supply curve is \(q=0.5 p-25\) and the demand curve is \(q=165-0.5 p\), where \(p\) is price in dollars. Find the equilibrium price and quantity before and after the tax is imposed.

Table \(1.20\) gives the sales, \(S\), of Intel Corporation, a leading manufacturer of integrated circuits. \({ }^{41}\) (a) Find the change in sales between 2003 and 2008 . (b) Find the average rate of change in sales between 2003 and 2008. Give units and interpret your answer. $$ \begin{array}{l} \text { Table } 1.20 \text { Intel sales, in millions of dollars }\\\ \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 \\ \hline S & 30,100 & 34,200 & 38,800 & 35,400 & 38,300 & 37,600 \\ \hline \end{array} \end{array} $$

A person is to be paid $$\$ 2000$$ for work done over a year. Three payment options are being considered. Option 1 is to pay the $$\$ 2000$$ in full now. Option 2 is to pay $$\$ 1000$$ now and $$\$ 1000$$ in a year. Option 3 is to pay the full $$\$ 2000$$ in a year. Assume an annual interest rate of \(5 \%\) a year, compounded continuously. (a) Without doing any calculations, which option is the best option financially for the worker? Explain. (b) Find the future value, in one year's time, of all three options. (c) Find the present value of all three options.

The population, \(P\), in millions, of Nicaragua was \(5.4\) million in 2004 and growing at an annual rate of \(3.4 \%\). Let \(t\) be time in years since 2004 . (a) Express \(P\) as a function in the form \(P=P_{0} a^{t}\). (b) Express \(P\) as an exponential function using base \(e\). (c) Compare the annual and continuous growth rates.

The demand and supply curves for a product are given in terms of price, \(p\), by $$ q=2500-20 p \quad \text { and } \quad q=10 p-500 $$ (a) Find the equilibrium price and quantity. Represent your answers on a graph. (b) A specific tax of $$\$ 6$$ per unit is imposed on suppliers. Find the new equilibrium price and quantity. Represent your answers on the graph. (c) How much of the $$\$ 6$$ tax is paid by consumers and how much by producers? (d) What is the total tax revenue received by the government?

A business associate who owes you $$\$ 3000$$ offers to pay you $$\$ 2800$$ now, or else pay you three yearly installments of $$\$ 1000$$ each, with the first installment paid now. If you use only financial reasons to make your decision, which option should you choose? Justify your answer, assuming a \(6 \%\) interest rate per year, compounded continuously.

What annual percent growth rate is equivalent to a continuous percent growth rate of \(8 \% ?\)

In Example 8 , the demand and supply curves are given by \(q=100-2 p\) and \(q=3 p-50\), respectively; the equilibrium price is $$\$ 30$$ and the equilibrium quantity is 40 units. A sales tax of \(5 \%\) is imposed on the consumer. (a) Find the equation of the new demand and supply curves. (b) Find the new equilibrium price and quantity. (c) How much is paid in taxes on each unit? How much of this is paid by the consumer and how much by the producer? (d) How much tax does the government collect?

Values of \(F(t), G(t)\), and \(H(t)\) are in Table 1.21. Which graph is concave up and which is concave down? Which function is linear? $$ \begin{array}{l} \text { Table } 1.21\\\ \begin{array}{c|c|c|c} \hline t & F(t) & G(t) & H(t) \\ \hline 10 & 15 & 15 & 15 \\ 20 & 22 & 18 & 17 \\ 30 & 28 & 21 & 20 \\ 40 & 33 & 24 & 24 \\ 50 & 37 & 27 & 29 \\ 60 & 40 & 30 & 35 \\ \hline \end{array} \end{array} $$

What continuous percent growth rate is equivalent to an annual percent growth rate of \(10 \%\) ?

Experiments suggest that the male maximum heart rate (the most times a male's heart can safely beat in a minute) decreases by 9 beats per minute during the first 21 years of his life, and by 26 beats per minute during the first 33 years \({ }^{43}\) If you model the maximum heart rate as a function of age, should you use a function that is increasing or decreasing, concave up or concave down?