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? \( \$ \) Round your answer to the nearest cent.

We will solve the problem in two steps: Step 1. Determine how much you need at retirement (in 25 years) so that you can withdraw $50,000 a year for 30 years. At retirement, the $50,000 withdrawals form an annuity with a 10% annual interest rate. The present value (at the retirement date) of an annuity that pays R dollars for n years at interest rate r is given by   PV = R × [1 – (1 + r)^(–n)] / r. Here, R = 50,000, n = 30, and r = 0.10. Thus, the amount needed at retirement is   PV_retirement = 50,000 × [1 – (1.1)^(–30)] / 0.10. Calculating the factor:   (1.1)^(30) ≈ 17.4494  ⇒  (1.1)^(–30) ≈ 1/17.4494 ≈ 0.0573.   Then 1 – 0.0573 ≈ 0.9427.   Divide by 0.10: 0.9427 / 0.10 = 9.427. Now multiply by 50,000:   PV_retirement ≈ 50,000 × 9.427 ≈ 471,350 dollars. Step 2. Find the annual deposit d over the next 25 years that will accumulate to this amount, given that the account earns 10% per year. When making annual deposits (at the end of each year), the future value of these deposits in 25 years is given by   FV = d × [(1 + r)^(n) – 1] / r. Here, FV = 471,350, r = 0.10, and n = 25. Substitute and solve for d:   d = FV × (r) / [(1.1)^(25) – 1]. First, calculate (1.1)^(25):   (1.1)^(25) ≈ 10.8347. So,   (1.1)^(25) – 1 ≈ 10.8347 – 1 = 9.8347, and dividing by r:   [(1.1)^(25) – 1] / 0.10 = 9.8347 / 0.10 = 98.347. Now, solving for d:   d ≈ 471,350 / 98.347 ≈ 4,792.30 dollars. Thus, you will need to deposit approximately $4,792.30 each year for 25 years to reach your retirement goal.