You want to be able to withdraw \( \$ 30,000 \) each year for 30 years. Your account earns \( 10 \% \) interest. a) How much do you need in your account at the beginning? b) How much total money will you pull out of the account? \( \$ 900,000 \) c) How much of that money is interest? \( \$ \) \$

**Step 1. Present Value of an Annuity Formula** We want the account to fund annual withdrawals of \( A = \$30{,}000 \) for \( n = 30 \) years at an annual interest rate of \( i = 10\% = 0.10 \). The present value (PV) needed is given by \[ PV = A \times \frac{1 - (1 + i)^{-n}}{i}. \] --- **Step 2. Substitute the Given Values** Substitute \( A = 30000 \), \( i = 0.10 \), and \( n = 30 \): \[ PV = 30000 \times \frac{1 - (1.1)^{-30}}{0.10}. \] --- **Step 3. Compute \( (1.1)^{-30} \)** First, compute \( (1.1)^{30} \). It is approximately \[ (1.1)^{30} \approx 17.45. \] Thus, \[ (1.1)^{-30} \approx \frac{1}{17.45} \approx 0.0573. \] --- **Step 4. Plug the Value Back into the Formula** Now substitute \( (1.1)^{-30} \approx 0.0573 \): \[ PV = 30000 \times \frac{1 - 0.0573}{0.10}. \] Simplify the numerator: \[ 1 - 0.0573 = 0.9427. \] Thus, \[ PV = 30000 \times \frac{0.9427}{0.10} = 30000 \times 9.427. \] --- **Step 5. Calculate the Present Value** Multiplying, \[ 30000 \times 9.427 \approx 282810. \] So, the amount needed in the account at the beginning is approximately \[ \$282{,}810. \] --- **Step 6. Total Money Withdrawn** Over 30 years, the total withdrawals amount to \[ 30000 \times 30 = \$900{,}000. \] --- **Step 7. Total Interest Earned** The interest earned is the difference between the total withdrawals and the initial deposit: \[ \text{Interest} = 900000 - 282810 \approx \$617{,}190. \] --- **Final Answers:** a) Approximately \( \$282{,}810 \) b) \( \$900{,}000 \) c) Approximately \( \$617{,}190 \) in interest