Question

Exponential Functions: Suppose your great-great-grandfather invested $400 earning 3.5% interest compounded continuously 200 years ago. How much would his investment be worth today?

a) $122,301.38
b) $221,289.04
c) $50,318.08
d) $980,234.67

Answer 1

To find how much a $400 investment earning 3.5% interest compounded continuously for 200 years is worth today, you use the continuous compounding interest formula A = Pe^rt. The calculation gives a result of approximately $122,301.38.

Step-by-step explanation:

The question involves using the formula for continuous compounding interest to determine the future value of an investment. The formula for continuous compounding is A = Pert, where P is the principal amount (the initial amount of money), r is the annual interest rate (expressed as a decimal), t is the time the money is invested for, and e is the base of the natural logarithm, approximately equal to 2.71828. To solve the problem:

1. Convert the interest rate from a percentage to a decimal by dividing by 100: 3.5%/100 = 0.035.
2. Use the continuous compounding interest formula with P=$400, r=0.035, and t=200.
3. Calculate the investment's worth today: A = 400e(0.035)(200).

After performing the calculation, you find that after 200 years, the investment would be worth approximately $122,301.38, which corresponds to option (a) among the choices provided.