Question 12 of 15 , step 1 of 1 Andrew deposits a fixed monthly amount into an annuity account for his child's college fund. He wishes to accumulate a future value of \( \$ 65,000 \) in 14 years. Assuming an APR of \( 3.6 \% \) compounded monthly, how much of the \( \$ 65,000 \) will Andrew ultimately deposit in the account, and how much is interest earned? Round your answers to the nearest cent, if necessary. Formulas

To determine how much Andrew will ultimately deposit and how much interest he will earn, we can start with the future value of an annuity formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] where: - \( FV \) is the future value ($65,000), - \( P \) is the monthly deposit, - \( r \) is the monthly interest rate (APR/12), - \( n \) is the total number of deposits (months). Given the APR of \( 3.6\% \) compounded monthly, the monthly interest rate \( r = \frac{0.036}{12} = 0.003 \). Over 14 years, \( n = 14 \times 12 = 168 \) months. Now, rearranging the formula to solve for \( P \): \[ P = \frac{FV \times r}{(1 + r)^n - 1} \] Plugging in the values, we calculate \( P \): \[ P = \frac{65000 \times 0.003}{(1 + 0.003)^{168} - 1} \] Calculating \( (1 + 0.003)^{168} \) first gives approximately \( 1.747422 \). Then we find: \[ P = \frac{65000 \times 0.003}{1.747422 - 1} \approx \frac{195}{0.747422} \approx 260.56 \] So, Andrew would deposit approximately $260.56 per month. Over 168 months, the total deposits he makes would be: \[ Total \ Deposits = P \times n \approx 260.56 \times 168 \approx 43,831.68 \] The interest earned is then: \[ Interest = FV - Total \ Deposits \approx 65,000 - 43,831.68 \approx 21,168.32 \] Thus, the final amounts are: - Total Deposited: \( \$43,831.68 \) - Interest Earned: \( \$21,168.32 \)