You want to be able to withdraw \( \$ 20,000 \) from your account each year for 20 years after you retire, You expect to retire in 15 years. If your account earns \( 9 \% \) interest, how much will you need to deposit each year until retirement to achieve your retirement goals? 7231.96 Round your answer to the nearest cent. Question Help: D video D port to forum

To determine how much you need to deposit each year until retirement to achieve your retirement goals, we can use the formula for calculating the present value of an annuity: \[ PV = \frac{PMT}{r} \left( 1 - \frac{1}{(1 + r)^n} \right) \] Where: - \( PV \) is the present value (the amount you need to deposit each year until retirement), - \( PMT \) is the annual withdrawal amount (\$20,000), - \( r \) is the annual interest rate (9% or 0.09), - \( n \) is the number of years until retirement (15 years). Given that you want to withdraw \$20,000 each year for 20 years after retirement, we need to calculate the present value of the annuity for 15 years until retirement. Substitute the values into the formula: \[ PV = \frac{20000}{0.09} \left( 1 - \frac{1}{(1 + 0.09)^{15}} \right) \] Now, we can calculate the present value to determine how much you need to deposit each year until retirement. Calculate the value by following steps: - step0: Calculate: \(\frac{20000}{0.09}\times \left(1-\frac{1}{\left(1+0.09\right)^{15}}\right)\) - step1: Add the numbers: \(\frac{20000}{0.09}\times \left(1-\frac{1}{1.09^{15}}\right)\) - step2: Convert the expressions: \(\frac{20000}{0.09}\times \left(1-\frac{1}{\left(\frac{109}{100}\right)^{15}}\right)\) - step3: Simplify: \(\frac{20000}{0.09}\times \left(1-\frac{100^{15}}{109^{15}}\right)\) - step4: Subtract the numbers: \(\frac{20000}{0.09}\times \frac{109^{15}-100^{15}}{109^{15}}\) - step5: Divide the numbers: \(\frac{2000000}{9}\times \frac{109^{15}-100^{15}}{109^{15}}\) - step6: Multiply the fractions: \(\frac{2000000\left(109^{15}-100^{15}\right)}{9\times 109^{15}}\) - step7: Multiply: \(\frac{2000000\times 109^{15}-2000000\times 100^{15}}{9\times 109^{15}}\) The present value of the annuity is approximately \$161,213.77. Therefore, you need to deposit approximately \$16,121.38 each year until retirement to achieve your retirement goals.