Comparing monthly payments and total costs of two loans Milan is taking out an amortized loan for \( \$ 17,000 \) to buy a new car and is deciding between the offers from two lenders. He wants to know which one would be the better deal over the life of the car loan, and by how much. Answer each part. Do not round intermediate computations, and round your answers to the nearest cent. If necessary, refer to the list of financial formulas. (a) A bank has offered him a 5 -year car loan at an annual interest rate of \( 8.8 \% \). Find the monthly payment. \( s[] \) (b) An online lending company has offered him a 6 -year car loan at an annual interest rate of \( 9.1 \% \). Find the monthly payment. \( \$ \square \) (c) Suppose Milan pays the monthly payment each month for the full term. Which lender's car loan would have the lowest total amount to pay off, and by how much? Bank The total amount paid would be \( \$ \square \) less than to the online lending company. Online lending company The total amount paid would be \( \$ \square \) less than to the bank. ( \( \$ \).

**(a)** For an amortized loan the monthly payment is given by \[ \text{Payment}=\frac{P\cdot r}{1-(1+r)^{-n}}, \] where - \(P\) is the principal (here, \(17000\)), - \(r\) is the monthly interest rate, - \(n\) is the total number of payments. For the bank’s offer: - Annual rate is \(8.8\%\), so the monthly rate is \[ r=\frac{0.088}{12}\approx0.00733333. \] - The term is 5 years so the total number of payments is \[ n=5\times12=60. \] Thus the monthly payment is \[ \text{Payment}=\frac{17000\times0.00733333}{1-(1+0.00733333)^{-60}}. \] Evaluating the expression (without rounding intermediate steps) gives a payment of approximately \[ \boxed{\$350.99}. \] --- **(b)** For the online lending company’s offer: - Annual rate is \(9.1\%\) so the monthly rate is \[ r=\frac{0.091}{12}\approx0.00758333. \] - The term is 6 years so \[ n=6\times12=72. \] The monthly payment is then \[ \text{Payment}=\frac{17000\times0.00758333}{1-(1+0.00758333)^{-72}}. \] Evaluating this gives a monthly payment of approximately \[ \boxed{\$307.42}. \] --- **(c)** Next we find the total amount paid over the life of each loan. For the bank: \[ \text{Total}_{\text{bank}}=60\times350.99\approx\$21059.40. \] For the online lending company: \[ \text{Total}_{\text{online}}=72\times307.42\approx\$22134.24. \] The difference in the total amounts is \[ \text{Difference}=\$22134.24-\$21059.40\approx\$1074.84. \] Since the bank’s loan totals less, we have \[ \boxed{\text{The bank’s loan costs about \$1074.84 less than the online lending company’s loan.}} \]