To determine Katelyn's monthly payment for a \[tex]$10,000 loan at an annual interest rate of 5.6% over 5 years with monthly payments, we use the monthly payment formula:
\[ M = \frac{P \times \left( \frac{r}{n} \right) \times (1 + \frac{r}{n})^{n \times t}}{(1 + \frac{r}{n})^{n \times t} - 1} \]
where:
- \( P \) is the principal amount (\$[/tex]10,000),
- [tex]\( r \)[/tex] is the annual interest rate (0.056),
- [tex]\( t \)[/tex] is the number of years (5),
- [tex]\( n \)[/tex] is the number of payments per year (12).
First, we compute the monthly interest rate:
[tex]\[ \text{Monthly interest rate} = \frac{r}{n} = \frac{0.056}{12} = 0.004666666666666667 \][/tex]
Next, we find the total number of payments:
[tex]\[ \text{Total number of payments} = t \times n = 5 \times 12 = 60 \][/tex]
Now, we plug these values into the formula:
[tex]\[ M = \frac{10000 \times 0.004666666666666667 \times (1 + 0.004666666666666667)^{60}}{(1 + 0.004666666666666667)^{60} - 1} \][/tex]
After solving this expression, we find Katelyn's monthly payment to be:
[tex]\[ M = 191.47 \][/tex]
Hence, rounded to the nearest cent, Katelyn's monthly payment for the loan is \$191.47.
