Chapter 6

Find the present and future values of an income stream of \(\$ 3000\) per year over a 15 -year period, assuming a \(6 \%\) annual interest rate compounded continuously.

A town has a population of 1000 . Fill in the table assuming that the town's population grows by (a) 50 people per year (b) \(5 \%\) per year $$ \begin{array}{c|c|c|c|c|c|c|c} \hline \text { Year } & 0 & 1 & 2 & 3 & 4 & \ldots & 10 \\ \hline \text { Population } & 1000 & & & & & \ldots & \\ \hline \end{array} $$

Draw a graph, with time in years on the horizontal axis, of what an income stream might look like for a company that sells sunscreen in the northeast United States.

Birth and death rates are often reported as births or deaths per thousand members of the population. What is the relative rate of growth of a population with a birth rate of 30 births per 1000 and a death rate of 20 deaths per 1000 ?

Find the average value of \(g(t)=1+t\) over the interval \([0,2]\)

(a) Find the present and future value of an income stream of \(\$ 6000\) per year for a period of 10 years if the interest rate, compounded continuously, is \(5 \%\). (b) How much of the future value is from the income stream? How much is from interest?

Find the consumer surplus for the demand curve \(p=\) \(100-3 q^{2}\) when 5 units are sold.

Find the average value of the function \(f(x)=5+4 x-x^{2}\) between \(x=0\) and \(x=3\).

Given the demand curve \(p=35-q^{2}\) and the supply curve \(p=3+q^{2}\), find the producer surplus when the market is in equilibrium.

Find the present and future values of an income stream of \(\$ 12,000\) a year for 20 years. The interest rate is \(6 \%\) compounded continuously.

A small business expects an income stream of \(\$ 5000\) per year for a four- year period. (a) Find the present value of the business if the annual interest rate, compounded continuously, is (i) \(3 \%\) (ii) \(10 \%\) (b) In each case, find the value of the business at the end of the four-year period.

$$ \begin{array}{l} \text { Find the consumer surplus for the demand curve } p=\\\ 100-4 q \text { when } q=10 \end{array} $$

Table \(6.3\) shows the cumulative number of AIDS deaths worldwide. \({ }^{6}\) Find the absolute increase in AIDS deaths between 2003 and 2004 and between 2006 and 2007 . Find the relative increase between 2003 and 2004 and between 2006 and 2007 . $$ \begin{array}{c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 \\ \hline \text { Cases } & 30.2 & 33.3 & 35.5 & 37.6 & 39.6 \\ \hline \end{array} $$

A bond is guaranteed to pay \(100+10 t\) dollars per year for 10 years, where \(t\) is in years from the present. Find the present value of this income stream, given an interest rate of \(5 \%\), compounded continuously.

A population is 100 at time \(t=0\), with \(t\) in years. (a) If the population has a constant absolute growth rate of 10 people per year, find a formula for the size of the population at time \(t\) (b) If the population has a constant relative growth rate of \(10 \%\) per year, find a formula for the size of the population at time \(t\). (c) Graph both functions on the same axes.

A recently-installed machine earns the company revenue at a continuous rate of \(60,000 t+45,000\) dollars per year during the first six months of operation and at the continuous rate of 75,000 dollars per year after the first six months. The cost of the machine is \(\$ 150,000\), the interest rate is \(7 \%\) per year, compounded continuously, and \(t\) is time in years since the machine was installed. (a) Find the present value of the revenue earned by the machine during the first year of operation. (b) Find how long it will take for the machine to pay for itself; that is, how long it will take for the present value of the revenue to equal the cost of the machine?

The size of a bacteria population is 4000 . Find a formula for the size, \(P\), of the population \(t\) hours later if the population is decreasing by (a) 100 bacteria per hour (b) \(5 \%\) per hour In which case does the bacteria population reach 0 first?

At what constant, continuous rate must money be deposited into an account if the account is to contain \(\$ 20,000\) in 5 years? The account earns \(6 \%\) interest compounded continuously.

Your company needs \(\$ 500,000\) in two years' time for renovations and can earn \(9 \%\) interest on investments. (a) What is the present value of the renovations? (b) If your company deposits money continuously at a constant rate throughout the two-year period, at what rate should the money be deposited so that you have the \(\$ 500,000\) when you need it?

The demand curve for a product has equation \(p=\) \(20 e^{-0.002 q}\) and the supply curve has equation \(p=\) \(0.02 q+1\) for \(0 \leq q \leq 1000\), where \(q\) is quantity and \(p\) is price in \$/unit. (a) Which is higher, the price at which 300 units are supplied or the price at which 300 units are demanded? Find both prices. (b) Sketch the supply and demand curves. Find the equilibrium price and quantity. (c) Using the equilibrium price and quantity, calculate and interpret the consumer and producer surplus.

A company is expected to earn \(\$ 50,000\) a year, at a continuous rate, for 8 years. You can invest the earnings at an interest rate of \(7 \%\), compounded continuously. You have the chance to buy the rights to the earnings of the company now for \(\$ 350,000 .\) Should you buy? Explain.

Show graphically that the maximum total gains from trade occurs at the equilibrium price. Do this by showing that if outside forces keep the price artificially high of low, the total gains from trade (consumer surplus \(+\) producer surplus) are lower than at the equilibrium price.

The value, \(V\), of a Tiffany lamp, worth \(\$ 225\) in 1975, increases at \(15 \%\) per year. Its value in dollars \(t\) years after 1975 is given by $$ V=225(1.15)^{t} $$ Find the average value of the lamp over the period \(1975-\) \(2010 .\)

Sales of Version \(6.0\) of a computer software package start out high and decrease exponentially. At time \(t\), in years, the sales are \(s(t)=50 e^{-t}\) thousands of dollars per year. After two years, Version \(7.0\) of the software is released and replaces Version \(6.0\). You can invest earnings at an interest rate of \(6 \%\), compounded continuously. Calculate the total present value of sales of Version 6,0 over the two-year period.

Rent controls on apartments are an example of price controls on a commodity. They keep the price artificially low (below the equilibrium price). Sketch a graph of supply and demand curves, and label on it a price \(p^{-}\) below the equilibrium price. What effect does forcing the price down to \(p^{-}\) have on: (a) The producer surplus? (b) The consumer surplus? (c) The total gains from trade (Consumer surplus + Producer surplus)?

Intel Corporation is a leading manufacturer of integrated circuits. In 2004 , Intel generated profits at a continuous rate of \(7.5\) billion dollars per year. \({ }^{2}\) Assume the interest rate was \(8.5 \%\) per year compounded continuously. (a) What was the present value of Intel's profits over the 2004 one-year time period? (b) What was the value at the end of the year of Intel's profits over the 2004 one-year time period?

If \(t\) is measured in days since June 1 , the inventory \(I(t)\) for an item in a warehouse is given by $$ I(t)=5000(0.9)^{t} $$ (a) Find the average inventory in the warehouse during the 90 days after June 1 . (b) Graph \(I(t)\) and illustrate the average graphically.

Harley-Davidson Inc. manufactures motorcycles. During the years following 2003 (the company's \(100^{\text {th }}\) anniversary), the company's net revenue can he approximated \(^{3}\) by \(4.6+0.4 t\) billion dollars per year, where \(t\) is time in years since January 1,2003 . Assume this rate holds through January 1,2013, and assume a continuous interest rate of \(3.5 \%\) per year. (a) What was the net revenue of the Harley-Davidson Company in \(2003 ?\) What is the projected net revenue in \(2013 ?\) (b) What was the present value, on January 1,2003 , of Harley-Davidson's net revenue for the ten years from January 1,2003 to January \(1,2013 ?\) (c) What is the future value, on January 1,2013, of net revenue for the preceding 10 years?

The total gains from trade (consumer surplus \(+\) producer surplus) is largest at the equilibrium price. What about the consumer surplus and producer surplus separately? (a) Suppose a price is artificially high. Can the consumer surplus at the artificial price be larger than the consumer surplus at the equilibrium price? What about the producer surplus? Sketch possible supply and demand curves to illustrate your answers. (b) Suppose a price is artificially low. Can the consumer surplus at the artificial price be larger than the consumer surplus at the equilibrium price? What about the producer surplus? Sketch possible supply and demand curves to illustrate your answers.

The supply and demand curves have equations \(p=S(q)\) and \(p=D(q)\), respectively, with equilibrium at \(\left(q^{*}, p^{*}\right)\). Using Riemann sums, explain the economic significance of \(\int_{0}^{q^{*}} S(q) d q\) to the producers.

Your company is considering buying new production machinery. You want to know how long it will take for the machinery to pay for itself; that is, you want to find the length of time over which the present value of the profit generated by the new machinery equals the cost of the machinery. The new machinery costs \(\$ 130,000\) and earns profit at the continuous rate of \(\$ 80,000\) per year. Use an interest rate of \(8.5 \%\) per year compounded continuously.

The supply and demand curves have equations \(p=S(q)\) and \(p=D(q)\), respectively, with equilibrium at \(\left(q^{*}, p^{*}\right)\). Using Riemann sums, give an interpretation of producer surplus, \(\int_{0}^{q^{*}}\left(p^{*}-S(q)\right) d q\) analogous to the interpretation of consumer surplus.

An oil company discovered an oil reserve of 100 million barrels. For time \(t>0\), in years, the company's extraction plan is a linear declining function of time as follows: $$ q(t)=a-b t $$ where \(q(t)\) is the rate of extraction of oil in millions of barrels per year at time \(t\) and \(b=0.1\) and \(a=10\). (a) How long does it take to exhaust the entire reserve? (b) The oil price is a constant \(\$ 20\) per barrel, the extraction cost per barrel is a constant \(\$ 10\), and the market interest rate is \(10 \%\) per year, compounded continuously. What is the present value of the company's profit?

The population of the world \(t\) years after 2000 is predicted to be \(P=6.1 e^{0.0125 t}\) billion. (a) What population is predicted in 2010 ? (b) What is the predicted average population between 2000 and 2010 ?

The value of good wine increases with age. Thus, if you are a wine dealer, you have the problem of deciding whether to sell your wine now, at a price of \(\$ P\) a bottle, or to sell it later at a higher price. Suppose you know that the amount a wine-drinker is willing to pay for a bottle of this wine \(t\) years from now is \(\$ P(1+20 \sqrt{t})\). Assuming continuous compounding and a prevailing interest rate of \(5 \%\) per year, when is the best time to sell your wine?

The number of hours, \(H\), of daylight in Madrid as a function of date is approximated by the formula $$ H=12+2.4 \sin [0.0172(t-80)] $$ where \(t\) is the number of days since the start of the year. Find the average number of hours of daylight in Madrid: (a) in January (b) in June (c) over a year (d) Explain why the relative magnitudes of your answers to parts (a), (b), and (c) are reasonable.

The population, \(P\), in millions, of Nicaragua was \(5.78\) million in 2008 and growing at an annual rate of \(1.8 \%\). (a) Write a formula for \(P\) as a function of \(t\), where \(t\) is years since 2008 . (b) Find the projected average rate of change (or absolute growth rate) in Nicaragua between 2008 and 2009 , and between 2009 and \(2010 .\) Explain why your answers are different. (c) Use your answers to part (b) to confirm that the relative rate of change (or relative growth rate) over both time intervals was \(1.8 \%\).

A bar of metal is cooling from \(1000^{\circ} \mathrm{C}\) to room temperature, \(20^{\circ} \mathrm{C}\). The temperature, \(H\), of the bar \(t\) minutes after it starts cooling is given, in \({ }^{\circ} \mathrm{C}\), by $$ H=20+980 e^{-0.1 t} $$ (a) Find the temperature of the bar at the end of one hour. (b) Find the average value of the temperature over the first hour. (c) Is your answer to part (b) greater or smaller than the average of the temperatures at the beginning and the end of the hour? Explain this in terms of the concavity of the graph of \(H\).

The rate of sales (in sales per month) of a company is given, for \(t\) in months since January 1 , by $$ r(t)=t^{4}-20 t^{3}+118 t^{2}-180 t+200 . $$ (a) Graph the rate of sales per month during the first year \((t=0\) to \(t=12\) ). Does it appear that more sales were made during the first half of the year, or during the second half? (b) Estimate the total sales during the first 6 months of the year and during the last 6 months of the year. (c) What are the total sales for the entire year? (d) Find the average sales per month during the year.

Throughout much of the \(20^{\text {th }}\) century, the yearly consumption of electricity in the US increased exponentially at a continuous rate of \(7 \%\) per year. Assume this trend continues and that the electrical energy consumed in 1900 was \(1.4\) million megawatt-hours. (a) Write an expression for yearly electricity consumption as a function of time, \(t\), in years since \(1900 .\) (b) Find the average yearly electrical consumption throughout the \(20^{\text {th }}\) century. (c) During what year was electrical consumption closest to the average for the century? (d) Without doing the calculation for part (c), how could you have predicted which half of the century the answer would be in?