You want to be able to withdraw \( \$ 50,000 \) from your account each year for 30 years after you retire. You expect to retire in 25 years. If your account earns \( 10 \% \) interest, how much will you need to deposit each year until retirement to achieve your retirement goals? \( \$ 4792.30 \) Round your answer to the nearest cent.

We have two steps: Step 1. Determine the lump sum needed at retirement. You wish to withdraw $50,000 each year for 30 years (an annuity) at an annual interest rate of 10%. The present value of this annuity (at retirement) is given by   Lump Sum = 50,000 × [(1 − 1/1.1^30) / 0.10]. Calculating 1.1^30 (using a calculator) gives approximately 17.4494, so   (1 − 1/17.4494) ≈ 1 − 0.0573 = 0.9427,   Then, Lump Sum ≈ 50,000 × (0.9427/0.10) = 50,000 × 9.427 = 471,350. Step 2. Determine the annual deposit required from now until retirement. You plan to make annual deposits for 25 years into an account earning 10% per year. The future value of an annuity-immediate is   Future Value = Deposit × [(1.1^25 − 1) / 0.10]. You need this to equal the lump sum calculated in Step 1. Hence, set   Deposit × [(1.1^25 − 1) / 0.10] = 471,350. Solve for Deposit:   Deposit = 471,350 × [0.10 / (1.1^25 − 1)]. Using a calculator, 1.1^25 ≈ 10.834705 so that   (1.1^25 − 1) ≈ 9.834705. Then,   Deposit ≈ 471,350 × (0.10 / 9.834705)       ≈ 471,350 × 0.01017       ≈ $4792.30. Thus, you need to deposit approximately $4,792.30 each year for 25 years to meet your retirement goals.