Chapter 1

A company is considering whether to buy a new machine, which costs $$\$ 97,000$$. The cash flows (adjusted for taxes and depreciation) that would be generated by the new machine are given in the following table: $$ \begin{array}{c|c|c|c|c} \hline \text { Year } & 1 & 2 & 3 & 4 \\ \hline \text { Cash flow } & \$ 50,000 & \$ 40,000 & \$ 25,000 & \$ 20,000 \\ \hline \end{array} $$ (a) Find the total present value of the cash flows. Treat each year's cash flow as a lump sum at the end of the year and use an interest rate of \(7.5 \%\) per year, compounded annually. (b) Based on a comparison of the cost of the machine and the present value of the cash flows, would you recommend purchasing the machine?

The population of a city is 50,000 in 2008 and is growing at a continuous yearly rate of \(4.5 \%\). (a) Give the population of the city as a function of the number of years since 2008 . Sketch a graph of the population against time. (b) What will be the city's population in the year 2018 ? (c) Calculate the time for the population of the city to reach \(100,000 .\) This is called the doubling time of the population.

A car starts slowly and then speeds up. Eventually the car slows down and stops. Graph the distance that the car has traveled against time.

You win $$\$ 38,000$$ in the state lottery to be paid in two installments -$$\$ 19,000$$ now and $$\$ 19,000$$ one year from now. A friend offers you $$\$ 36,000$$ in return for your two lottery payments. Instead of accepting your friend's offer, you take out a one-year loan at an interest rate of \(8.25 \%\) per year, compounded annually. The loan will be paid back by a single payment of $$\$ 19,000$$ (your second lottery check) at the end of the year. Which is better, your friend's offer or the loan?

In 2000 , there were about 213 million vehicles (cars and trucks) and about 281 million people in the US. The number of vehicles has been growing at \(4 \%\) a year, while the population has been growing at \(1 \%\) a year. If the growth rates remain constant, when is there, on average, one vehicle per person?

You are considering whether to buy or lease a machine whose purchase price is $$\$ 12,000$$. Taxes on the machine will be $$\$ 580$$ due in one year, $$\$ 464$$ due in two years, and $$\$ 290$$ due in three years. If you buy the machine, you expect to be able to sell it after three years for $$\$ 5,000 .$$ If you lease the machine for three years, you make an initial payment of $$\$ 2650$$ and then three payments of $$\$ 2650$$ at the end of each of the next three years. The leasing company will pay the taxes. The interest rate is \(7.75 \%\) per year, compounded annually. Should you buy or lease the machine? Explain.

Which relative change is bigger in magnitude? Justify your answer. The change in the Dow Jones average from \(164.6\) to \(77.9\) in 1931 ; the change in the Dow Jones average from \(13261.8\) to \(8776.4\) in 2008

(a) Write an equation for a graph obtained by vertically stretching the graph of \(y=x^{2}\) by a factor of 2, followed by a vertical upward shift of 1 unit. Sketch it. (b) What is the equation if the order of the transformations (stretching and shifting) in part (a) is interchanged? (c) Are the two graphs the same? Explain the effect of reversing the order of transformations.

You are buying a car that comes with a one-year warranty and are considering whether to purchase an extended warranty for $$\$ 375 .$$ The extended warranty covers the two years immediately after the one-year warranty expires. You estimate that the yearly expenses that would have been covered by the extended warranty are $$\$ 150$$ at the end of the first year of the extension and $$\$ 250$$ at the end of the second year of the extension. The interest rate is \(5 \%\) per year, compounded annually. Should you buy the extended warranty? Explain.

Which relative change is bigger in magnitude? Justify your answer. The change in the US population from \(5.2\) million to \(7.2\) million from 1800 to 1810 ; the change in the US population from \(151.3\) to \(179.3\) from 1950 to 1960 .

Which relative change is bigger in magnitude? Justify your answer. An increase in class size from 5 to 10 ; an increase in class size from 30 to 50 .

Which relative change is bigger in magnitude? Justify your answer. An increase in sales from $$\$ 100,000$$ to $$\$ 500,000$$; an increase in sales from $$\$ 20,000,000$$ to $$\$ 20,500,000$$.

Find the relative change of a population if it changes (a) From 1000 to 2000 (b) From 2000 to 1000 (c) From \(1,000,000\) to \(1,001,000\)

On Black Monday, October 28,1929, the stock market on Wall Street crashed. The Dow Jones average dropped from \(298.94\) to \(260.64\) in one day. What was the relative change in the index?

On May 11,2009, the cost to mail a letter in the US was raised from 42 cents to 44 cents. Find the relative change in the cost.

The US Consumer Price Index (CPI) is a measure of the cost of living. The inflation rate is the annual relative rate of change of the CPI. Use the January data in Table \(1.22^{44}\) to estimate the inflation rate for each of years \(2005-2008\). $$ \begin{array}{l} \text { Table } 1.22\\\ \begin{array}{c|c|c|c|c|c} \hline \text { Year } & 2005 & 2006 & 2007 & 2008 & 2009 \\ \hline \text { CPI } & 190.7 & 198.3 & 202.416 & 211.08 & 211.143 \\ \hline \end{array} \end{array} $$

During 2008 the US economy stopped growing and began to shrink. Table \(1.23^{45}\) gives quarterly data on the US Gross Domestic Product (GDP) which measures the size of the economy. (a) Estimate the relative growth rate (percent per year) at the first four times in the table. (b) Economists often say an economy is in recession if the GDP decreases for two quarters in a row. Was the US in recession in 2008 ? $$ \begin{array}{l} \text { Table } 1.23\\\ \begin{array}{c|c|c|c|c|c} \hline t \text { (years since 2008) } & 0 & 0.25 & 0.5 & 0.75 & 1.0 \\ \hline \text { GDP (trillion dollars) } & 14.15 & 14.29 & 14.41 & 14.2 & 14.09 \\\ \hline \end{array} \end{array} $$