Final answer:
Grayson needs to invest approximately $12,276 to reach $15,000 in an account with a 4% continuously compounded interest rate over 5 years, according to the formula for continuous compounding.
Step-by-step explanation:
To calculate how much Grayson needs to initially invest for the value of the account to reach $15,000 in 5 years at a 4% interest rate compounded continuously, we will 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.
- and e is the base of the natural logarithm, approximately 2.71828.
In this example, we want to solve for P when A is $15,000, r is 0.04 (since the interest rate is 4%), and t is 5 years. Rearranging the formula to solve for P gives us P = A/ert.
Plugging the numbers in, we get:
P = $15,000 / e(0.04)(5)
Calculating the value inside the exponent first:
e(0.04)(5) ≈ e0.20 ≈ 1.22140275
Now we divide:
P ≈ $15,000 / 1.22140275 ≈ $12,276.04
Therefore, to the nearest dollar, Grayson needs to initially invest $12,276.