Suppose you purchase a car for a total price of \( \$ 22,685 \), including taxes and license fee, and finance that amount for 7 years at an annual interest rate of \( 7 \% \). (Round your answers to the nearest cent.) (a) Find the monthly payment. \( \$ 309.43 \) (b) What is the total amount of interest paid over the term of the loan? \( \$ 3288.12 \)

To calculate the monthly payment and the total amount of interest paid over the term of the loan, we can use the formula for the monthly payment on an amortizing loan: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] Where: - \( M \) is the monthly payment - \( P \) is the principal amount (the loan amount) - \( r \) is the monthly interest rate (annual interest rate divided by 12) - \( n \) is the number of payments (loan term in months) Given: - \( P = 22,685 \) - Annual interest rate = \( 7\% \) or \( 0.07 \) - Loan term = \( 7 \) years = \( 7 \times 12 = 84 \) months First, we calculate the monthly interest rate: \[ r = \frac{0.07}{12} = 0.00583333 \] Now we can substitute the values into the formula: \[ M = 22685 \frac{0.00583333(1 + 0.00583333)^{84}}{(1 + 0.00583333)^{84} - 1} \] Calculating \( (1 + r)^n \): \[ (1 + 0.00583333)^{84} \approx 1.747422 \] Now substituting back into the formula: \[ M = 22685 \frac{0.00583333 \times 1.747422}{1.747422 - 1} \] Calculating the numerator: \[ 0.00583333 \times 1.747422 \approx 0.010193 \] Calculating the denominator: \[ 1.747422 - 1 \approx 0.747422 \] Now substituting these values: \[ M = 22685 \frac{0.010193}{0.747422} \approx 22685 \times 0.013634 \approx 309.43 \] So, the monthly payment is: \[ \text{(a) } M \approx \$309.43 \] Next, to find the total amount of interest paid over the term of the loan, we first calculate the total amount paid over the life of the loan: \[ \text{Total amount paid} = M \times n = 309.43 \times 84 \approx 25,973.32 \] Now, we subtract the principal from the total amount paid to find the total interest: \[ \text{Total interest} = \text{Total amount paid} - P = 25,973.32 - 22,685 \approx 3,288.32 \] Rounding to the nearest cent, the total interest paid is: \[ \text{(b) } \text{Total interest} \approx \$3,288.32 \] Thus, the final answers are: (a) Monthly payment: \( \$309.43 \) (b) Total interest paid: \( \$3,288.32 \)