Problem One thousand dollars is invested at 5% interest compounded continuously.

2. Problem • One thousand dollars is invested at 5% interest compounded continuously. • Give the formula for A(t), the compounded amount after t years. • How much will be in the account after 6 years? • After 6 years, at what rate will A(t) be growing? • How long is required to double the initial investment?

3. Problem Pablo Picasso’s “The Dream” was purchased in 1941 for a war distressed price of $7000. The painting was sold in 1997 for $48.4 million, the second highest price ever paid for a Picasso painting at auction. What rate of interest compounded continuously did the investment earn?

4. Retirement Income Historically, investments in the stock market have yielded an average rate of 11.7% per year (over the long haul). Suppose on graduating high school a rich aunt deposits 10,000 in an account at an 11% annual interest rate that compounds continuously. She claims that you will have over a million dollars by retirement time (age 65). Is she right? 65 – 18 = 47 years A = 10,000e0.11·47 = 10,000e5.17 = 1,759,148.38 over 1.75 million Yes!

5. Savings Account Problems If you deposit $540 in an account that pays 9.1% interest compounded continuously what will your balance be in 20 years? Compounded continuously!!! A = Pert

6. Continuous growth • The population is a certain city increases by 2.3% per year. Assuming the population in 2000 was 7.6 million, what is the population expected to be in 2010? What was the population in 1980? • A certain radioactive compound decays at a rate of 0.0345% per year. Given 200 grams of the compound, how much will remain after 500 years? After 5000 years? How about 500 years ago? • Your bank is offering a savings account with a nominal rate of 1.5%, compounded continuously. If you deposit $1,000 in 2010, what will your balance be in 2020? What is the effective annual yield?

7. Continuously compounded • If you invest $1,000 at an annual interest rate of 5% compounded continuously, calculate the final amount you will have in the account after five years. • If you invest $500 at an annual interest rate of 10% compounded continuously, calculate the final amount you will have in the account after five years. • If you invest $2,000 at an annual interest rate of 13% compounded continuously, calculate the final amount you will have in the account after 20 years. • If you invest $20,000 at an annual interest rate of 1% compounded continuously, calculate the final amount you will have in the account after 20 years.

8. Solve the following using the formula for compounding continuously. Show all work for credit. • 1) a) If Columbus invested 1¢ ($0.01) at 2% interest with a Native American banker when he landed in the 'New • World', • how much would be in the account today rounded to the nearest cent? • 0.01e^((2001-14920)*0.02) • 1a) $263.70 • b) If Columbus invested 1¢ ($0.01) at 4% interest with a Native American banker when he landed in the 'New • World', • how much would be in the account today rounded to the nearest cent? • 0.01e^((2001-14920)*0.02) • 1b) $6954015.46 • c) If Columbus invested 1¢ ($0.01) with a Native American banker when he landed in the 'New World', • at what interest percent rate would he need to find to have a billion dollars today? • (Note: Write the answer as a percent rounded to 2 decimal places.) • ln(1000000000/0.01)/(2001-1492) • 1c) 4.98 % • 2) If today you invested 50¢ ($0.50) at 4.5% interest, how many years will it take to accumulate to $1,000,000? • (Round to the nearest tenth of a year.) • ln(1000000/0.5)/(0.045) • 2) 322.4 years