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? \( \$ 900,000 \)

We start with the present value formula for an annuity: \[ PV = \frac{W \left( 1 - (1+r)^{-n} \right)}{r} \] where \( W = 30000 \) (annual withdrawal), \( r = 0.1 \) (annual interest rate), and \( n = 30 \) (number of years). **Step 1. Plug in the values:** \[ PV = \frac{30000 \left( 1 - (1.1)^{-30} \right)}{0.1} \] **Step 2. Compute \((1.1)^{-30}\):** \[ (1.1)^{-30} = \frac{1}{(1.1)^{30}} \] Using the logarithmic approximation: \[ (1.1)^{30} \approx e^{30 \ln(1.1)} \approx e^{30 \times 0.09531} \approx e^{2.8593} \approx 17.45 \] Thus, \[ (1.1)^{-30} \approx \frac{1}{17.45} \approx 0.0573 \] **Step 3. Evaluate the term in the parentheses:** \[ 1 - (1.1)^{-30} \approx 1 - 0.0573 = 0.9427 \] **Step 4. Calculate the present value:** \[ PV \approx \frac{30000 \times 0.9427}{0.1} = 30000 \times 9.427 \approx 282810 \] Rounding to the nearest dollar, you need approximately \(\$282,810\) in your account at the beginning. **Step 5. Compute the total amount withdrawn over 30 years:** \[ \text{Total Withdrawn} = 30000 \times 30 = 900000 \] --- **Answers:** a) Approximately \(\$282,810\) required at the beginning. b) A total of \(\$900,000\) will be withdrawn over the 30 years.