Eddy purchased a club membership costing \( \$ 2,530 \). He made a down payment of \( \$ 530 \) and financed the balance with an installment loan for 48 months. If the payments are \( \$ 59.28 \) each month, use Table \( 13-1 \) to find the APR. Select one: o a. \( 19.75 \% \) o b. \( 16.00 \% \) o c. \( 18.50 \% \) o d. \( 17.50 \% \)

1. Compute the amount financed. The club membership costs \$2,530 and Eddy made a down payment of \$530: \[ \text{Amount financed} = 2530 - 530 = 2000. \] 2. Use the formula for the present value of an annuity: \[ PV = A \times \frac{1 - (1+i)^{-n}}{i}, \] where - \( PV = 2000 \), - \( A = 59.28 \) (monthly payment), - \( n = 48 \) months, - \( i \) is the monthly interest rate. Thus, we have: \[ 2000 = 59.28 \times \frac{1 - (1+i)^{-48}}{i}. \] 3. Divide both sides by 59.28: \[ \frac{2000}{59.28} = \frac{1 - (1+i)^{-48}}{i}. \] Calculating the left-hand side: \[ \frac{2000}{59.28} \approx 33.73. \] This gives the equation: \[ \frac{1 - (1+i)^{-48}}{i} \approx 33.73. \] 4. Solve for \( i \) by trying a trial value. A starting guess is \( i = 0.015 \) (1.5% per month): - Calculate \((1.015)^{48}\). Using the approximation: \[ (1.015)^{48} \approx e^{48 \ln(1.015)} \approx e^{48 \times 0.014889} \approx e^{0.715} \approx 2.044. \] - Then, \[ (1.015)^{-48} \approx \frac{1}{2.044} \approx 0.489. \] - Compute the annuity factor: \[ \frac{1 - 0.489}{0.015} \approx \frac{0.511}{0.015} \approx 34.07. \] This value is slightly above 33.73, meaning \( i \) needs to be a bit larger. 5. Try \( i = 0.0155 \) (1.55% per month): - Calculate \((1.0155)^{48}\). Approximate using: \[ (1.0155)^{48} \approx e^{48 \ln(1.0155)} \quad \text{with } \ln(1.0155) \approx 0.0154, \] \[ 48 \times 0.0154 \approx 0.7392, \quad e^{0.7392} \approx 2.094. \] - Then, \[ (1.0155)^{-48} \approx \frac{1}{2.094} \approx 0.4775. \] - Compute the annuity factor: \[ \frac{1 - 0.4775}{0.0155} \approx \frac{0.5225}{0.0155} \approx 33.71. \] This is nearly equal to 33.73, confirming that \( i \approx 0.0155 \) (1.55% per month). 6. Convert the monthly interest rate to an annual percentage rate (APR) by multiplying by 12: \[ \text{APR} \approx 0.0155 \times 12 = 0.186 \text{ or } 18.6\%. \] 7. Comparing with the choices: - a. \( 19.75\% \) - b. \( 16.00\% \) - c. \( 18.50\% \) - d. \( 17.50\% \) The result \( 18.6\% \) is closest to \( 18.50\% \). Thus, the correct answer is: \( \textbf{c. } 18.50\% \).