Activity 6.1

The Genie's Offer: Exponential Growth and Linear Growth

10 points

Due at the beginning of class, Friday, February 27, 2009

In this activity you will explore differences between exponential and linear growth. You will also analyze an example where value is decreasing exponentially.

1. Suppose a magic genie offers you a choice. The genie will give you $1000 on the first day of the year. On each succeeding day of January, the genie will give you $1000 more than what you received on the previous day, until the end of the month. Or you may choose to receive 2 cents on Jan. 1, and then each day for the rest of the month, the genie will give you double the amount you had received on the previous day.

1. a. Which deal sounds better to you and why?

1. b. Set up a spreadsheet to complete the following table. Use the appropriate formulas for the second and third columns, and fill in to the end of January.

(As a hint to help you fill in the spreadsheet, remember that you can enter numbers 1 and 2 in the first two cells in a column and then drag down to fill in the rest of the integers. You can also enter a formula and drag it down a column and then use Audit mode to check the formulas entered. See Actvity 4.1 if you need to refresh your memory. Remember that Excel's auto-fill can only figure out linear patterns.) Paste this spreadsheet into your Word document when it is finished.

1. c. What does the table show?

1. d. At what point in the table is the amount in the third column greater than the amount in the second column and what does that mean?

1. e. Looking at your spreadsheet, identify a pattern and write an equation that shows how much money m you will have after d days if you take the genie's first offer of $1000 on January 1 and an additional $1000 each day after that.

1. f. Identify the types of functions you wrote in part (e) of this question. How do you know they are those types of function?

1. g. Now you'll examine the genie's second offer, which is to give you 2 cents on Jan. 1 and then on each succeeding day to double the amount you had the day before.

Analyze the pattern emerging below, and then develop an equation relating how much money m you will have after d days if you take the genie's second offer (and enter this answer in your Word document). When you type your answer, you may use the ^ symbol for exponentiation.

Money, in dollars, on day 1 = original amount of $ = 0.02 = 0.02
Money, in dollars, on day 2 = $ on Jan. 2 = ($ on Jan. 1) * 2 = 0.02 * 2 = 0.02 * 2¹
Money, in dollars, on day 3 = $ on Jan. 3 = ($ on Jan. 2) * 2 = 0.02 * 2 * 2 = 0.02 * 2²
Money, in dollars, on day 4 = $ on Jan. 4 = ($ on Jan. 3) * 2 = 0.02 * 2 * 2 * 2 = 0.02 * 2³
Money, in dollars, on day 5 = $ on Jan. 5 = ($ on Jan. 4) * 2 = 0.02 * 2 * 2 * 2 * 2 = 0.02 * 2^4
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Money, in dollars, on day d is given by: m_d = _____________________________ (What you fill in should be an expression in terms of d).

1. h. Fill in the blanks:
1. h. i. On Jan. 20, you have ______________ times as much money as you had on Jan. 1.

1. h. ii. On Jan. 31, you have ______________ times as much money as you had on Jan. 1.

1. i. Explain why it makes sense to call the function you wrote in part (g) of the question an exponential function and the kind of growth seen in the genie's second offer exponential growth.

1. j. In Excel, create a SINGLE graph which shows the amount of money you would have for each of the genie's offers. You should only graph these functions (which give the amount of money) for the first 25 days. The days will be your explanatory variable. What does your graph show?

2. Appliances decrease in value as soon as you take them out of the store. A certain appliance originally cost $1,100. After one year the appliance is worth $935. Assume that the decline in value is exponential; that is, assume that the ratio of the appliance's value in one year to the appliance's value in the previous year is constant.

2. a. Calculate the ratio (value_in_year_1) / (value_in_year_0).
Note that this can also be written as value_in_year_1 = ratio * value_in_year_0 , and that this ratio will be constant for all succeeding pairs of years.

2. b. Set up a spreadsheet, starting with the following table, to compute the value of the appliance for years 2 through 20. You will need to enter the appropriate formula to compute the value of the appliance in year 2 and then drag it down the column for succeeding years. You should then paste this table into your Word document.

2. c. What is the appliance's value after 10 years?

2. d. Approximately when is the appliance's value half of its original value?

2. e. Approximately when is the appliance's value one-quarter of its original value?

2. f. If you continue these assumptions, will the appliance ever be worth $0? Explain.

2. g. Use the letters v (which is equal to the appliance's value, in $) and y (which is equal to the # of years from the purchase date), and write a formula for v in terms of y. (Note that your formula must be set up so that when y = 0, v = 1100 , when y = 1, v = 935 , and so on.)

2. h. Is v an exponential function of y? Why or why not?

2. i. Describe the similarities and differences between the formula for v in part (h) of this question and the formula for m in #1 (g).

Summary
In this activity, you practiced creating formulas and scatterplots in Excel. You compared linear and exponential growth and explored the difference between these two fundamental types of growth. You also looked at a quantity that decreases exponentially and investigated patterns in several types of growth to find appropriate formulas.