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 \( \$ 85,000 \) in 13 years. Assuming an APR of \( 3.8 \% \) compounded monthly, how much of the \( \$ 85,000 \) will Andrew ultimately deposit in the account, and how much is interest earned? Round your answers to the nearest cent. if necessary. Formulas

To solve Andrew's scenario, we can use the future value of an annuity formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) is the future value, - \( P \) is the amount deposited each month, - \( r \) is the monthly interest rate (annual rate divided by 12), and - \( n \) is the total number of payments (months). Here, we know \( FV = 85,000 \), the APR is \( 3.8\% \) (which gives us a monthly interest rate of \( \frac{0.038}{12} \approx 0.00316667 \)), and \( n = 13 \times 12 = 156 \). 1. Rearranging the formula to find \( P \): \[ P = \frac{FV \times r}{(1 + r)^n - 1} \] Substituting the known values: - \( FV = 85,000 \) - \( r = 0.00316667 \) - \( n = 156 \) Calculating \( P \): \[ P = \frac{85000 \times 0.00316667}{(1 + 0.00316667)^{156} - 1} \] Calculating the denominator: \[ (1 + 0.00316667)^{156} \approx 1.682288 \] So, \[ P \approx \frac{85000 \times 0.00316667}{1.682288 - 1} \approx \frac{269.167}{0.682288} \approx 395.11 \] Thus, Andrew needs to deposit approximately \( P \approx 395.11 \) monthly. 2. To find the total deposited amount: \[ Total\ Deposited = P \times n = 395.11 \times 156 \approx 61,819.16 \] 3. To find the interest earned: \[ Interest\ Earned = Future\ Value - Total\ Deposited = 85,000 - 61,819.16 \approx 23,180.84 \] In conclusion, Andrew will ultimately deposit approximately \( \$ 61,819.16 \) into the account, and he will earn about \( \$ 23,180.84 \) in interest. Now, for some extra context: Did you know that annuities have been a popular choice for long-term savings and investment since the Roman Empire? Back then, people utilized similar principles to secure their financial futures! In today's world, investing through annuities is a favored strategy for parents like Andrew. Many folks use these accounts not just for education savings but also for retirement planning, giving them peace of mind knowing their funds will grow over time.