Activity 7.2

Estimation, Scientific Notation, and Properties of Logarithms

10 points

Due at the beginning of class, Wednesday, March 4, 2009

In this activity, you will use scientific notation to develop an estimate of a large quantity by breaking it down into small pieces. You will also use properties of logarithms to investigate and answer questions about an investment.

1. One of Ross Perot's policy recommendations in his 1992 Presidential campaign was a call for a $0.50 tax on every gallon of gasoline sold in the U.S. Since that time, other lawmakers have supported additional gas taxes to help reduce consumption and raise money. This activity asks you to determine roughly how much revenue a $0.50 per gallon tax on gasoline would generate in a year. Before doing any calculations, take a guess as to how much revenue this proposal would generate in a year.

2. A useful strategy for estimating a quantity such as this one is to separate the problem into its component pieces and then to estimate each piece separately. These estimates can and should be fairly rough and are useful to provide a sense of the magnitude of a quantity.

2. a. What is the current population of the U.S.? (You can guess or check out the http://www.census.gov/ web-site for an up-to-date reading). Write this estimate in scientific notation.

2. b. Based on your answer to #2.(a), estimate the number of automobiles in the U.S. Write this estimate in scientific notation.

2. c. Estimate the number of miles traveled by a typical car in the U.S., in a year. Write this estimate in scientific notation.

2. d. Multiply your answers to #2(b) and #2(c) to obtain an estimate of the number of miles driven in the U.S. in one year.
Remember that when you multiply numbers in scientfic notation, you add the exponents:
2 x 10² * 4 x 10⁵ = 8 x 10⁷
and when you divide numbers in scientific notation, you subtract exponents:
(4 x 10⁵) / (2 x 10²) = 2 x 10³

Write this result in scientific notation.

2. e. Estimate the number of miles that a typical car travels per gallon of gasoline.

2. f. Divide your answers in #2(d) by your answer in #2(e) to estimate the number of gallons of gasoline purchased in the U.S. in one year. Write this result in scientific notation.

2. g. Use the answer above to estimate the revenue that would be generated by Perot's $0.50 per gallon tax on gasoline.

2. h. Is this a valid estimate? What do you think would happen to gasoline consumption if there was a sudden price increase? What happened to gasoline consumption when prices grew dramatically? How would you modify your estimate?

3. Now you will investigate an equation that relates to compound interest. Suppose you deposit $1,000 in a savings account that pays 5% annual interest, where the interest is compounded annually. Assuming you don't withdraw any of the money or interest, after one year your savings account will have
1,000 + 0.05*1,000 = 1,000 + 50 = $1,050.00.

How much will you have after two years? After three years?

4. Set up an Excel worksheet to help calculate the money in the account after a period of time, using the following instructions:

4. a. Open a new sheet in your Excel workbook and label column A as years after initial deposit and column B as amount in the account.

4. b. Enter numbers 0 through 20 in column A, starting in cell A2, and the corresponding amounts in column B. Use Excel to do the computations for you by entering a recursive formula in B3 that gives the amount in B3 in terms of the amount already in B2, and then drag to fill in the rest of the column.

Copy and paste this table into your Word document.

5. What formula did you enter in cell B3?

6. What amount did you obtain for the account balance after 20 years?

7. The account balance after n years (still assuming an initial depost of $1,000 and that no money is withdrawn during those years) can be given by a single formula in terms of n:

Amount after n years = 1000 * (1.05)ⁿ

In column C of your spreadsheet, enter the values obtained using this formula. (To do this, place your cursor in cell C2 and enter the appropriate formula, =1000*(1.05)^A2. Then drag down to year 20.) Paste this table into your Word document.

8. How do the entries in columns B and C compare, and why?

9. Using the previous formula, you can find the account balance after any number of years, without using a spreadsheet. You can also calculate the number of years it would take for the account balance to reach any given amount. For example, to find out how many years it would take for the account to reach $10,000, you need to find the value of n so that 10,000 = 1000 * (1.05)ⁿ .

Why is this equation the one we need to solve?

10. To solve for the number of years it would take to read $10,000 in the account, we need to get n out of the exponent. If we apply the logarithm function to both sides of the equation, we get the following:

log( 10,000 ) = log( 1,000 ) + n*log( 1.05 )

What properties of logarithms did we use to get this equation?

11. Because log( 10,000 ) = 4 and log( 1,000 ) = 3, the previous equation is equivalent to

4 = 3 + n*log( 1.05 ) .

Explain why log( 10,000 ) = 4 and log( 1,000 ) = 3.

12. Use algebra to solve (by hand, no calculations yet), for n in terms of log(1.05), and show your steps in your Word document.

13. Use a calculator or Excel to calculate an approximate value of log(1.05), and use it to approximate n, which is the number of years it would take for the account to reach $10,000.

n = _______________

14. Use the same approach that you used to above, in #7 - #13, to find the number of years it would take the account to reach $100,000, and show your steps in your Word document.

15. Drag down in your Excel sheet to verify your answers to #13 and #14 above (Do NOT include this table in your Word document). Do your answers above agree with the answers Excel calculated?

Summary
In this activity, you used scientific notation to make a rough estimate of the revenue that would be generated by a $0.50 tax per gallon of gasoline. You used Excel to calculate compound interest and used logarithms to solve equations to determine how many years a savings account investment should be kept in the bank to obtain a certain return.