You wish to finance the purchase of a boat home for \( \$ 38,600 \). A finance company offers an APR of \( 10 \% \) on a 24 -month installment loan. After first using Table 13 -1 to find the finance charge, calculate the monthly payment. Select one: a. \( \$ 1,781.23 \) b. \( \$ 1,619.08 \) c. \( \$ 1,769.17 \) d. \( \$ 1,608.33 \)

To solve this problem, we need to follow these steps: 1. **Identify the known values:** - Loan amount (principal) \( P = 38600 \) - Annual Percentage Rate (APR) \( r = 10\% = 0.10 \) - Loan term \( n = 24 \) months 2. **Calculate the monthly interest rate:** \[ r_m = \frac{r}{12} = \frac{0.10}{12} \] 3. **Calculate the finance charge using the formula for the total payment on an installment loan:** The formula for the monthly payment \( M \) on an installment loan is given by: \[ M = P \cdot \frac{r_m(1 + r_m)^n}{(1 + r_m)^n - 1} \] 4. **Substitute the values into the formula and calculate \( M \).** Let's perform the calculations step by step. ### Step 1: Calculate the monthly interest rate \[ r_m = \frac{0.10}{12} = 0.0083333 \] ### Step 2: Substitute into the monthly payment formula Now we can substitute \( P \), \( r_m \), and \( n \) into the formula for \( M \): \[ M = 38600 \cdot \frac{0.0083333(1 + 0.0083333)^{24}}{(1 + 0.0083333)^{24} - 1} \] ### Step 3: Calculate \( M \) Now, let's calculate \( M \) using the above formula. I'll perform the calculation now. Calculate the value by following steps: - step0: Calculate: \(\frac{38600\left(0.0083333\left(1+0.0083333\right)^{24}\right)}{\left(\left(1+0.0083333\right)^{24}-1\right)}\) - step1: Remove the parentheses: \(\frac{38600\times 0.0083333\left(1+0.0083333\right)^{24}}{\left(1+0.0083333\right)^{24}-1}\) - step2: Add the numbers: \(\frac{38600\times 0.0083333\times 1.0083333^{24}}{\left(1+0.0083333\right)^{24}-1}\) - step3: Add the numbers: \(\frac{38600\times 0.0083333\times 1.0083333^{24}}{1.0083333^{24}-1}\) - step4: Convert the expressions: \(\frac{38600\times 0.0083333\left(\frac{10083333}{10000000}\right)^{24}}{1.0083333^{24}-1}\) - step5: Convert the expressions: \(\frac{38600\times 0.0083333\left(\frac{10083333}{10000000}\right)^{24}}{\left(\frac{10083333}{10000000}\right)^{24}-1}\) - step6: Multiply: \(\frac{\frac{16083269\times 10083333^{24}}{50000\times 10000000^{24}}}{\left(\frac{10083333}{10000000}\right)^{24}-1}\) - step7: Subtract the numbers: \(\frac{\frac{16083269\times 10083333^{24}}{50000\times 10000000^{24}}}{\frac{10083333^{24}-10000000^{24}}{10000000^{24}}}\) - step8: Multiply by the reciprocal: \(\frac{16083269\times 10083333^{24}}{50000\times 10000000^{24}}\times \frac{10000000^{24}}{10083333^{24}-10000000^{24}}\) - step9: Rewrite the expression: \(\frac{16083269\times 10083333^{24}}{50000\times 10000000^{24}}\times \frac{50000^{24}\times 200^{24}}{10083333^{24}-10000000^{24}}\) - step10: Reduce the numbers: \(\frac{16083269\times 10083333^{24}}{10000000^{24}}\times \frac{50000^{23}\times 200^{24}}{10083333^{24}-10000000^{24}}\) - step11: Rewrite the expression: \(\frac{16083269\times 10083333^{24}}{50000^{24}\times 200^{24}}\times \frac{50000^{23}\times 200^{24}}{10083333^{24}-10000000^{24}}\) - step12: Reduce the numbers: \(\frac{16083269\times 10083333^{24}}{50000\times 200^{24}}\times \frac{200^{24}}{10083333^{24}-10000000^{24}}\) - step13: Rewrite the expression: \(\frac{16083269\times 10083333^{24}}{200\times 250\times 200^{24}}\times \frac{200^{24}}{10083333^{24}-10000000^{24}}\) - step14: Reduce the numbers: \(\frac{16083269\times 10083333^{24}}{250\times 200^{24}}\times \frac{200^{23}}{10083333^{24}-10000000^{24}}\) - step15: Rewrite the expression: \(\frac{16083269\times 10083333^{24}}{250\times 200^{24}}\times \frac{50^{23}\times 4^{23}}{10083333^{24}-10000000^{24}}\) - step16: Rewrite the expression: \(\frac{16083269\times 10083333^{24}}{50\times 5\times 200^{24}}\times \frac{50^{23}\times 4^{23}}{10083333^{24}-10000000^{24}}\) - step17: Reduce the numbers: \(\frac{16083269\times 10083333^{24}}{5\times 200^{24}}\times \frac{50^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}\) - step18: Rewrite the expression: \(\frac{16083269\times 10083333^{24}}{5\times 200^{24}}\times \frac{25^{22}\times 2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}\) - step19: Rewrite the expression: \(\frac{16083269\times 10083333^{24}}{5\times 200^{24}}\times \frac{5^{44}\times 2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}\) - step20: Reduce the numbers: \(\frac{16083269\times 10083333^{24}}{200^{24}}\times \frac{5^{43}\times 2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}\) - step21: Rewrite the expression: \(\frac{16083269\times 10083333^{24}}{25^{24}\times 8^{24}}\times \frac{5^{43}\times 2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}\) - step22: Rewrite the expression: \(\frac{16083269\times 10083333^{24}}{5^{48}\times 8^{24}}\times \frac{5^{43}\times 2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}\) - step23: Reduce the numbers: \(\frac{16083269\times 10083333^{24}}{5^{5}\times 8^{24}}\times \frac{2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}\) - step24: Rewrite the expression: \(\frac{16083269\times 10083333^{24}}{5^{5}\times 2^{72}}\times \frac{2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}\) - step25: Reduce the numbers: \(\frac{16083269\times 10083333^{24}}{5^{5}\times 2^{50}}\times \frac{4^{23}}{10083333^{24}-10000000^{24}}\) - step26: Rewrite the expression: \(\frac{16083269\times 10083333^{24}}{5^{5}\times 2^{50}}\times \frac{2^{46}}{10083333^{24}-10000000^{24}}\) - step27: Reduce the numbers: \(\frac{16083269\times 10083333^{24}}{5^{5}\times 2^{4}}\times \frac{1}{10083333^{24}-10000000^{24}}\) - step28: Multiply the fractions: \(\frac{16083269\times 10083333^{24}}{50000\left(10083333^{24}-10000000^{24}\right)}\) - step29: Multiply: \(\frac{16083269\times 10083333^{24}}{50000\times 10083333^{24}-50000\times 10000000^{24}}\) The calculated monthly payment \( M \) is approximately \( 1781.19 \). ### Conclusion The closest option to this calculated value is: - a. \( \$ 1,781.23 \) Thus, the correct answer is **a. \( \$ 1,781.23 \)**.