When Diane bought her condominium, she got her mortgage through her credit union. The mortgage was a personal, amortized loan for \( \$ 91,500 \), at an interest rate of \( 3.75 \% \), with monthly payments for a term of 40 years. For each part, do not round any intermediate computations and round your final answers to the nearest cent. If necessary, refer to the list of financial formulas. (a) Find Dlane's monthly payment. (b) If Diane pays the monthly payment each month for the full term, find her total amount to repay the loan. \( 5 \square \) (c) If Diane pays the monthly payment each month for the full term, find the total amount of interest she will pay. s

To solve the problem, we will use the formula for the monthly payment on an amortized loan, which is given by: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] where: - \( M \) = monthly payment - \( P \) = principal amount (loan amount) - \( r \) = monthly interest rate (annual interest rate divided by 12) - \( n \) = total number of payments (loan term in months) ### Given Data: - \( P = 91500 \) - Annual interest rate = \( 3.75\% \) or \( 0.0375 \) - Monthly interest rate \( r = \frac{0.0375}{12} \) - Loan term = \( 40 \) years or \( 40 \times 12 = 480 \) months Now, let's calculate the monthly payment \( M \). 1. Calculate the monthly interest rate \( r \): \[ r = \frac{0.0375}{12} \] 2. Calculate the total number of payments \( n \): \[ n = 40 \times 12 \] 3. Substitute \( P \), \( r \), and \( n \) into the monthly payment formula. Let's perform these calculations step by step. ### Step 1: Calculate \( r \) and \( n \) \[ r = \frac{0.0375}{12} \] \[ n = 40 \times 12 \] ### Step 2: Calculate the monthly payment \( M \) \[ M = 91500 \frac{r(1 + r)^n}{(1 + r)^n - 1} \] Now, I will perform these calculations. Calculate the value by following steps: - step0: Calculate: \(40\times 12\) - step1: Multiply the numbers: \(480\) Calculate or simplify the expression \( 0.0375/12 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.0375}{12}\) - step1: Convert the expressions: \(\frac{\frac{3}{80}}{12}\) - step2: Multiply by the reciprocal: \(\frac{3}{80}\times \frac{1}{12}\) - step3: Reduce the numbers: \(\frac{1}{80}\times \frac{1}{4}\) - step4: Multiply the fractions: \(\frac{1}{80\times 4}\) - step5: Multiply: \(\frac{1}{320}\) Calculate or simplify the expression \( 91500*(0.003125*(1+0.003125)^480)/((1+0.003125)^480-1) \). Calculate the value by following steps: - step0: Calculate: \(\frac{91500\left(0.003125\left(1+0.003125\right)^{480}\right)}{\left(\left(1+0.003125\right)^{480}-1\right)}\) - step1: Remove the parentheses: \(\frac{91500\times 0.003125\left(1+0.003125\right)^{480}}{\left(1+0.003125\right)^{480}-1}\) - step2: Add the numbers: \(\frac{91500\times 0.003125\times 1.003125^{480}}{\left(1+0.003125\right)^{480}-1}\) - step3: Add the numbers: \(\frac{91500\times 0.003125\times 1.003125^{480}}{1.003125^{480}-1}\) - step4: Convert the expressions: \(\frac{91500\times 0.003125\left(\frac{321}{320}\right)^{480}}{1.003125^{480}-1}\) - step5: Convert the expressions: \(\frac{91500\times 0.003125\left(\frac{321}{320}\right)^{480}}{\left(\frac{321}{320}\right)^{480}-1}\) - step6: Multiply: \(\frac{\frac{183\times 321^{480}}{2^{2884}\times 5^{478}}}{\left(\frac{321}{320}\right)^{480}-1}\) - step7: Subtract the numbers: \(\frac{\frac{183\times 321^{480}}{2^{2884}\times 5^{478}}}{\frac{321^{480}-320^{480}}{320^{480}}}\) - step8: Multiply by the reciprocal: \(\frac{183\times 321^{480}}{2^{2884}\times 5^{478}}\times \frac{320^{480}}{321^{480}-320^{480}}\) - step9: Rewrite the expression: \(\frac{183\times 321^{480}}{2^{2884}\times 5^{478}}\times \frac{64^{480}\times 5^{480}}{321^{480}-320^{480}}\) - step10: Rewrite the expression: \(\frac{183\times 321^{480}}{2^{2884}\times 5^{478}}\times \frac{2^{2880}\times 5^{480}}{321^{480}-320^{480}}\) - step11: Reduce the numbers: \(\frac{183\times 321^{480}}{2^{4}}\times \frac{5^{2}}{321^{480}-320^{480}}\) - step12: Multiply the fractions: \(\frac{183\times 321^{480}\times 5^{2}}{2^{4}\left(321^{480}-320^{480}\right)}\) - step13: Multiply: \(\frac{4575\times 321^{480}}{16\times 321^{480}-16\times 320^{480}}\) ### Step 1: Calculate \( r \) and \( n \) - The monthly interest rate \( r \) is approximately \( 0.003125 \). - The total number of payments \( n \) is \( 480 \). ### Step 2: Calculate the monthly payment \( M \) Using the formula for the monthly payment: \[ M = 91500 \frac{0.003125(1 + 0.003125)^{480}}{(1 + 0.003125)^{480} - 1} \] Calculating \( M \): \[ M \approx 368.31 \] ### (a) Monthly Payment Diane's monthly payment is approximately **$368.31**. ### (b) Total Amount to Repay the Loan To find the total amount to repay the loan, we multiply the monthly payment by the total number of payments: \[ \text{Total Repayment} = M \times n \] Calculating this: \[ \text{Total Repayment} = 368.31 \times 480 \] Let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(368.31\times 480\) - step1: Multiply the numbers: \(176788.8\) ### (b) Total Amount to Repay the Loan The total amount to repay the loan is: \[ \text{Total Repayment} \approx 176788.80 \] Thus, Diane's total amount to repay the loan is approximately **$176,788.80**. ### (c) Total Amount of Interest Paid To find the total amount of interest paid, we subtract the principal from the total repayment: \[ \text{Total Interest} = \text{Total Repayment} - P \] Calculating this: \[ \text{Total Interest} = 176788.80 - 91500 \] Let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(176788.8-91500\) - step1: Subtract the numbers: \(85288.8\) ### (c) Total Amount of Interest Paid The total amount of interest paid is: \[ \text{Total Interest} \approx 85288.80 \] Thus, Diane's total amount of interest she will pay is approximately **$85,288.80**. ### Summary of Results: - (a) Monthly Payment: **$368.31** - (b) Total Amount to Repay the Loan: **$176,788.80** - (c) Total Amount of Interest Paid: **$85,288.80**