Final answer:
The correct answer is C. $27,731.59. To solve for the principal amount (P) given the future value (A), interest rate (r), and time (t), we can use A = Pe^(rt) and rearrange it to P = A / (e^(rt)). After plugging in the values and calculating, we find that $27,731.59 needs to be invested at a 2.3% interest rate compounded continuously for 17 years to reach $41,000.
Step-by-step explanation:
To determine how much money must be invested at a 2.3% interest rate compounded continuously to reach $41,000 after 17 years, we can use the formula for continuous compounding, which is:
A = Pert
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- t is the time the money is invested for, in years.
- e is the base of the natural logarithm (approximately equal to 2.71828).
In this case, we want to solve for P, knowing that A = $41,000, r = 2.3%, or 0.023 (in decimal), t = 17 years, and e is the mathematical constant. Rearranging the formula to solve for P gives:
P = A / (ert)
Now we can plug in the values:
P = 41000 / (e(0.023 * 17))
P = 41000 / (e0.391)
P = 41000 / (e0.391)
Using a calculator, we find that:
P ≈ 41000 / (1.47896)
P ≈ 27731.59
The correct answer is C. $27,731.59, which is the amount that needs to be invested at a 2.3% interest rate compounded continuously to have $41,000 after 17 years.