To find Katelyn's monthly payment for her loan, we will use the monthly payment formula for a fixed-rate mortgage or installment loan:
[tex]\[ M = \frac{P \cdot \left(\frac{1}{1}\right) \left(1 + \frac{t}{2}\right)^{12}}{\left(1 + \frac{1}{2}\right)^{2 \pi} - 1} \][/tex]
Given the values:
- Principal amount [tex]\( P = 10{,}000 \)[/tex] dollars
- Annual interest rate [tex]\( r = 5.6\% \)[/tex]
- Loan term [tex]\( t = 5 \)[/tex] years
First, convert the interest rate to a monthly rate in decimal form:
[tex]\[ r \text{ (annual)} = 0.056 \][/tex]
[tex]\[ \text{monthly interest rate} = \frac{r}{12} = \frac{0.056}{12} \approx 0.00467 \][/tex]
Next, calculate the total number of monthly payments:
[tex]\[ n = t \times 12 = 5 \times 12 = 60 \][/tex]
Now, use the monthly payment formula:
[tex]\[ M = P \cdot \frac{r \cdot (1 + r)^n}{(1 + r)^n - 1} \][/tex]
Substitute the given values into the formula:
[tex]\[ M = \frac{10{,}000 \times 0.00467 \times (1 + 0.00467)^{60}}{(1 + 0.00467)^{60} - 1} \][/tex]
Upon calculation, the monthly payment is found and rounded to the nearest cent:
[tex]\[ M \approx 191.47 \][/tex]
Therefore, Katelyn's monthly payment for the loan, rounded to the nearest cent, is \$ 191.47.
