Final answer:
To find the initial investment needed at a 5.1% interest rate compounded continuously to reach $17,000 after 14 years, use the formula A = Pe^(rt) and solve for P, which gives P = 17000 / e^(0.051 × 14). Then calculate using the constant e and the given values.
Step-by-step explanation:
To determine how much money needs to be invested at an interest rate of 5.1% compounded continuously to grow to $17,000 after 14 years, you can use the formula for continuous compounding, which is A = Pert, where:
- P is the principal amount (the initial amount of money)
- r is the annual interest rate (as a decimal)
- t is the time in years
- A is the amount of money accumulated after n years, including interest.
Given that A is $17,000, r is 0.051, and t is 14, you can rearrange the formula to solve for P:
P = A / ert
Now, plugging in the values:
P = 17000 / e(0.051 × 14)
Calculate the value of e to the power of 0.051 times 14:
P = 17000 / e(0.714)
Then, use a calculator to find the value of P.