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Two cities each have a population of 1.2 million people. City A is growing by a factor of 1.15 every 10 years, while city \(\mathbf{B}\) is decaying by a factor of 0.85 every 10 years. a. Write an exponential function for each city's population \(P_{A}(t)\) and \(P_{B}(t)\) after \(t\) years. b. For each city's population function generate a table of values for \(x=0\) to \(x=50,\) using 10 -year intervals, then sketch a graph of each town's population on the same grid.

Mute swans were imported from Europe in the nineteenth century to grace ponds. Now there is concern that their population is growing too rapidly, edging out native species. Their population along the Atlantic coast has grown from 5800 in 1986 to 14,313 in 2002 . The increase is most acute in the mid-Atlantic region, but Massachusetts has also seen a jump, with 2939 mute swans counted in 2002 as compared with 585 in 1986 . a. Compare the growth factor in the mute swan population for the entire Atlantic coast with that for Massachusetts. b. Compare the average rate of change in the mute swan population for the entire Atlantic coast with that for Massachusetts. c. Construct both a linear and an exponential model for the mute swan population in Massachusetts since 1986 . d. Compare the projected populations of mute swans in Massachusetts by the year 2010 as predicted by your linear and exponential models.

Which function has the steepest graph? $$ \begin{array}{l} F(x)=100(1.2)^{x} \\ G(x)=100(0.8)^{x} \\ H(x)=100(1.2)^{-x} \end{array} $$

Estimate the doubling time using the rule of 70 when: a. \(P=2.1(1.0475)^{t}\), where \(t\) is in years b. \(Q=2.1(1.00475)^{T}\), where \(T\) is in years

(Graphing program recommended.) On the same graph, sketch \(f(x)=3(1.5)^{x}, g(x)=-3(1.5)^{x},\) and \(h(x)=3(1.5)^{-x}\) a. Which graphs are mirror images of each other across the \(y\) -axis? b. Which graphs are mirror images of each other across the \(x\) -axis? c. Which graphs are mirror images of each other about the origin (i.e., you could translate one into the other by reflecting first about the \(y\) -axis, then about the \(x\) -axis)? d. What can you conclude about the graphs of the two functions \(f(x)=C a^{x}\) and \(g(x)=-C a^{x} ?\) e. What can you conclude about the graphs of the two functions \(f(x)=C a^{x}\) and \(g(x)=C a^{-x} ?\)

[Part (e) requires use of the Internet and technology to find a best-fit function.] A "rule of thumb" used by car dealers is that the trade-in value of a car decreases by \(30 \%\) each year. a. Is this decline linear or exponential? b. Construct a function that would express the value of the car as a function of years owned. c. Suppose you purchase a car for \(\$ 15,000 .\) What would its value be after 2 years? d. Explain how many years it would take for the car in part (c) to be worth less than \(\$ 1000\). Explain how you arrived at your answer. e. Internet search: Go to the Internet site for the Kelley Blue Book (www.kbb.com). i. Enter the information about your current car or a car you would like to own. Specify the actual age and mileage of the car. What is the Blue Book value? ii. Keeping everything else the same, assume the car is I year older and increase the mileage by 10,000 . What is the new value? iii. Find a best-fit exponential function to model the value of your car as a function of years owned. What is the annual decay rate? iv. According to this function, what will the value of your car be 5 years from now?

In a chain letter one person writes a letter to a number of other people, \(N,\) who are each requested to send the letter to \(N\) other people, and so on. In a simple case with \(N=2\), let's assume person Al starts the process. Al sends to \(\mathrm{B} 1\) and \(\mathrm{B} 2 ; \mathrm{B} 1\) sends to \(\mathrm{C} 1\) and \(\mathrm{C} 2 ; \mathrm{B} 2\) sends to \(\mathrm{C} 3\) and \(\mathrm{C} 4\); and so on. A typical letter has listed in order the chain of senders who sent the letters. So \(\mathrm{D} 7\) receives a letter that has \(\mathrm{A} 1, \mathrm{~B} 2\), and \(\mathrm{C} 4\) listed. If these letters request money, they are illegal. A typical request looks like this: \(\cdot\) When you receive this letter, send \(\$ 10\) to the person on the top of the list. \(\cdot\) Copy this letter, but add your name to the bottom of the list and leave off the name at the top of the list. \(\cdot\) Send a copy to two friends within 3 days. For this problem, assume that all of the above conditions hold. a. Construct a mathematical model for the number of new people receiving letters at each level \(L,\) assuming \(N=2\) as shown in the above tree. b. If the chain is not broken, how much money should an individual receive? c. Suppose A 1 sent out letters with two additional phony names on the list (say Ala and Alb) with P.O. box addresses she owns. So both \(\mathrm{B} 1\) and \(\mathrm{B} 2\) would receive a letter with the list \(\mathrm{A} 1, \mathrm{~A} 1 \mathrm{a},\) Alb. If the chain isn't broken, how much money would Al receive? d. If the chain continued as described in part (a), how many new people would receive letters at level \(25 ?\) e. Internet search: Chain letters are an example of a "pyramid growth" scheme. A similar business strategy is multilevel marketing. This marketing method uses the customers to sell the product by giving them a financial incentive to promote the product to potential customers or potential salespeople for the product. (See Exercise \(31 .)\) Sometimes the distinction between multilevel marketing and chain letters gets blurred. Search the U.S. Postal Service website (www.usps.gov) for "pyramid schemes" to find information about what is legal and what is not. Report what you find.