Let's break down each part of the question step-by-step. --- ### **1. How much has accumulated in Hardaway’s deferred compensation account?** #### **Given:** - Annual deposit (\(PMT\)): $8,000 - Number of years (\(n\)): 30 - Interest rate (\(r\)): 7% - Deposits are made at **end of each year** → **Ordinary Annuity** We need the FV of an ordinary annuity: \[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \] \[ FV = 8,000 \times \frac{(1 + 0.07)^{30} - 1}{0.07} \] \[ (1 + 0.07)^{30} = (1.07)^{30} \approx 7.612255 \] \[ \frac{7.612255 - 1}{0.07} = \frac{6.612255}{0.07} \approx 94.4608 \] \[ FV = 8,000 \times 94.4608 \approx \$755,686 \] **Rounded Answer:** \[ \boxed{\$755,686} \] --- ### **2. How much will Hardaway be able to withdraw at the beginning of each year if he elects to receive payments for 16 years?** - Amount available: \$755,686 - Number of years (\(n\)): 16 - Interest rate (\(r\)): 7% - Withdrawals at **beginning** of each year → **Annuity Due** We need to solve for the annual payment of an annuity due (\(PMT\)) where present value = \$755,686. \[ PV = PMT \times \left[\frac{1 - (1 + r)^{-n}}{r}\right] \times (1 + r) \] \[ 755,686 = PMT \times \left[\frac{1 - (1.07)^{-16}}{0.07}\right] \times 1.07 \] First, calculate \((1.07)^{-16}\): \[ (1.07)^{16} \approx 2.969 \] \[ (1.07)^{-16} = \frac{1}{2.969} \approx 0.3368 \] \[ 1 - 0.3368 = 0.6632 \] \[ \frac{0.6632}{0.07} = 9.474 \] \[ 9.474 \times 1.07 = 10.137 \] So, \[ 755,686 = PMT \times 10.137 \] \[ PMT = \frac{755,686}{10.137} \approx \$74,573 \] **Rounded Answer:** \[ \boxed{74,573} \] --- ### **3. How many years will Hardaway be able to receive payments if he chooses to receive \$85,000 per year at the beginning of each year?** Now, \(PMT = 85,000\), \(PV = 755,686\), \(r = 7\%\), \(n = ?\), and it's an **annuity due**. \[ PV = PMT \times \left[\frac{1 - (1 + r)^{-n}}{r}\right] \times (1 + r) \] \[ 755,686 = 85,000 \times \left[\frac{1 - (1.07)^{-n}}{0.07}\right] \times 1.07 \] First, calculate \(85,000 \times 1.07 = 90,950\). \[ 755,686 = 90,950 \times \left[\frac{1 - (1.07)^{-n}}{0.07}\right] \] \[ \frac{755,686}{90,950} = \frac{1 - (1.07)^{-n}}{0.07} \] \[ 8.314 = \frac{1 - (1.07)^{-n}}{0.07} \] \[ 8.314 \times 0.07 = 1 - (1.07)^{-n} \] \[ 0.582 = 1 - (1.07)^{-n} \] \[ (1.07)^{-n} = 1 - 0.582 = 0.418 \] Now, solve for \(n\): \[ (1.07)^{-n} = 0.418 \] Take natural log of both sides: \[ -n \ln(1.07) = \ln(0.418) \] \[ -n \times 0.06766 = -0.872 \] \[ n = \frac{0.872}{0.06766} \approx 12.89 \] **Rounded Answer:** \[ \boxed{13} \] --- ## **Final Answers (rounded to nearest whole number):** 1. **Accumulated amount:** \$755,686 2. **Annual withdrawal for 16 years:** \$74,573 3. **Number of years at \$85,000/year:** 13 --- **If you need to see the formulas or steps in Excel or with tables, let me know!**