Sinal Exam = Spring 2024 Suppose that at age 30, you decide to save for retirement by depositing \( \$ 200 \) at the end of every month in an IRA that pays \( 3.5 \% \) compounded monthly. How much will you have from the IRA when you retire at age 65 ? Sohe problem. Round to the nearest dollar. O. \( \$ 102,687 \) OB. \( \$ 158,273 \) OC. \( \$ 164,441 \) OD. \( \$ 95,714 \)

To determine how much you'll have in your IRA when you retire at age 65, 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 of the annuity. - \( P \) is the amount deposited each period (monthly). - \( r \) is the interest rate per period. - \( n \) is the total number of deposits. Given: - \( P = 200 \) (monthly deposit), - Annual interest rate = \( 3.5\% \), so monthly interest rate \( r = \frac{0.035}{12} = 0.00291667 \), - You will save from age 30 to 65, which is \( 65 - 30 = 35 \) years, and since deposits are monthly, \( n = 35 \times 12 = 420 \). Now, we plug these values into the formula: \[ FV = 200 \times \frac{(1 + 0.00291667)^{420} - 1}{0.00291667} \] Calculating \( (1 + 0.00291667)^{420} \): \[ (1 + 0.00291667)^{420} \approx 3.32708 \] Now substituting back into our formula: \[ FV = 200 \times \frac{3.32708 - 1}{0.00291667} \] \[ FV \approx 200 \times \frac{2.32708}{0.00291667} \] \[ FV \approx 200 \times 797.198 \approx 159439.49 \] Rounding to the nearest dollar gives us \( \$ 159,439 \). However, among the options provided, \( \$ 158,273 \) is the closest, and thus, the answer is: **B. \( \$ 158,273 \)**.