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? c) How much of that money is interest? \( \$ \)

We are given: - Annual withdrawal: \( R = \$30,\!000 \) - Number of years: \( n = 30 \) - Annual interest rate: \( i = 10\% = 0.10 \) **(a) Find the present value (the amount needed at the beginning) of this annuity.** The present value of an annuity is given by \[ PV = R \cdot \frac{1 - (1+i)^{-n}}{i} \] Substitute the given values: \[ PV = 30000 \cdot \frac{1 - (1.10)^{-30}}{0.10} \] First, calculate \((1.10)^{-30}\): \[ (1.10)^{-30} = \frac{1}{(1.10)^{30}} \] A typical evaluation shows that \[ (1.10)^{30} \approx 17.4494 \quad \Longrightarrow \quad (1.10)^{-30} \approx 0.0573 \] Now substitute back: \[ PV \approx 30000 \cdot \frac{1 - 0.0573}{0.10} = 30000 \cdot \frac{0.9427}{0.10} = 30000 \cdot 9.427 \] \[ PV \approx 282810 \] Thus, you would need approximately \( \$282,\!810 \) at the beginning. --- **(b) Find the total amount of money withdrawn over the 30 years.** The total withdrawn is simply: \[ \text{Total Withdrawn} = R \times n = 30000 \times 30 = 900000 \] So, you would pull out a total of \( \$900,\!000 \). --- **(c) Determine how much of that money is interest.** The interest earned is the total withdrawn minus the initial amount: \[ \text{Interest} = \text{Total Withdrawn} - PV = 900000 - 282810 \approx 617190 \] Thus, approximately \( \$617,\!190 \) is interest.