Suppose that you borrow \( \$ 12,000 \) for five years at \( 5 \% \) toward the purchase of a car. Use PMT \( =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \) to find the monthly payments and the total interest for the loan. The monthly payment is \( \$ \square \). (Do not round until the final answer. Then round to the nearest cent as needed.)

To find the monthly payments for your loan using the given formula, we substitute the respective values into the equation. Here, \( P = 12000 \), \( r = 0.05 \) (annual interest rate), \( n = 12 \) (number of payments per year), and \( t = 5 \) (loan term in years). The monthly payment \( PMT \) can be calculated as follows: \[ PMT = \frac{12000\left(\frac{0.05}{12}\right)}{\left[1-\left(1+\frac{0.05}{12}\right)^{-12 \cdot 5}\right]} \] Calculating this step-by-step: 1. Calculate \( \frac{r}{n} = \frac{0.05}{12} \approx 0.00416667 \). 2. Calculate \( n \cdot t = 12 \cdot 5 = 60 \). 3. Compute \( \left(1 + \frac{r}{n}\right)^{-n \cdot t} = \left(1 + 0.00416667\right)^{-60} \approx \left(1.00416667\right)^{-60} \approx 0.779202 \). 4. Now compute \( 1 - 0.779202 \approx 0.220798 \). 5. Finally plug everything back into the formula: \[ PMT \approx \frac{12000 \cdot 0.00416667}{0.220798} \approx \frac{50}{0.220798} \approx 226.500 \] So, the monthly payment will be approximately \( \$ 226.50 \). Next, to find the total interest paid over the course of the loan, we first calculate the total payments made: \[ \text{Total Payments} = PMT \cdot n \cdot t = 226.500 \cdot 60 \approx 13590 \] Now, subtract the principal from the total payments to find the interest: \[ \text{Total Interest} = \text{Total Payments} - P = 13590 - 12000 = 1590 \] Thus, the monthly payment is \( \$ 226.50 \) and the total interest over five years is \( \$ 1590.00 \).