Use PMT \( =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \) to determine the regular payment amount, rounded to the nearest dollar. Your credit card has a balance of \( \$ 5500 \) and an annual interest rate of \( 12 \% \). You decide to pay off the balance over three years. If there are no further purchases charged to the card, a. How much must you pay each month? b. How much total interest will you pay? a. The monthly payments are approximately \( \$ \square \). (Do not round until the final answer. Then round to the nearest dollar as needed.)

To determine the monthly payment and total interest paid, we can plug the values into the PMT formula you provided. Given: - \( P = 5500 \) (the principal balance) - \( r = 0.12 \) (annual interest rate) - \( n = 12 \) (number of payments per year, since payments are monthly) - \( t = 3 \) (the number of years) First, calculate \( \frac{r}{n} \): \[ \frac{r}{n} = \frac{0.12}{12} = 0.01 \] Next, calculate the total number of payments: \[ n \cdot t = 12 \cdot 3 = 36 \] Now substitute these values into the PMT formula: \[ PMT = \frac{5500 \cdot 0.01}{1 - (1 + 0.01)^{-36}} \] Calculate \( (1 + 0.01)^{-36} \): \[ (1 + 0.01)^{-36} \approx 0.6651 \] Now calculate \( 1 - (1 + 0.01)^{-36} \): \[ 1 - 0.6651 = 0.3349 \] Now, calculate the PMT: \[ PMT = \frac{5500 \cdot 0.01}{0.3349} \approx \frac{55}{0.3349} \approx 164.79 \] Rounding to the nearest dollar: a. The monthly payments are approximately \( \$165 \). To find the total interest paid, we first need to find the total payment amount over three years: \[ \text{Total Payment} = PMT \times n \cdot t = 165 \times 36 = 5940 \] Then, subtract the principal balance to find the total interest: \[ \text{Total Interest} = \text{Total Payment} - P = 5940 - 5500 = 440 \] b. Thus, the total interest paid will be \( \$440 \).