Question

16 years ago, Crosby opened an account that compounds

interest on a continuous basis with an initial deposit of
$18,151. If the account currently has a balance $63,899,
what was the rate associated with the account?

Answer 1

To find the annual interest rate for an account that compounds interest continuously, we use the formula A = Pe^(rt). For an initial deposit of $18,151 that grew to $63,899 in 16 years, the rate is approximately 7.87%.

Step-by-step explanation:

To find the rate associated with the account that compounds interest continuously, we use the formula 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 (in decimal), t is the time the money is invested for, and e is Euler's number (approximately 2.71828).

Here, A is $63,899, P is $18,151, and t is 16 years. We need to solve for r, the interest rate. The formula is then written as:

63899 = 18151e(r)(16)

We divide both sides by 18151 to isolate e to the power of r multiplied by t:

e(r)(16) = 63899 / 18151
Next, we take the natural logarithm of both sides to solve for r times t:

ln(e(r)(16)) = ln(63899 / 18151)
(r)(16) = ln(63899 / 18151)
r = (ln(63899 / 18151)) / 16

After doing the calculations, we have:

r = (ln(3.5213)) / 16
r ≈ (1.259) / 16
r ≈ 0.0787 or 7.87%

Therefore, the annual interest rate associated with the account is approximately 7.87%.