To find the amount owed after 13 years when [tex]$28,000 is borrowed at an annual interest rate of 6.5% compounded annually, follow these steps:
1. Identify the given values:
- Principal amount (P): $[/tex]28,000
- Annual interest rate (r): 6.5% or 0.065 in decimal form
- Number of years (t): 13
2. Understand the formula for compound interest:
The formula to calculate the final amount [tex]\( A \)[/tex] when interest is compounded annually is:
[tex]\[
A = P \left(1 + r\right)^t
\][/tex]
where:
- [tex]\( A \)[/tex] is the final amount.
- [tex]\( P \)[/tex] is the principal amount.
- [tex]\( r \)[/tex] is the annual interest rate in decimal.
- [tex]\( t \)[/tex] is the number of years.
3. Plug in the given values into the formula:
[tex]\[
A = 28000 \left(1 + 0.065\right)^{13}
\][/tex]
4. Calculate the expression inside the parenthesis:
[tex]\[
1 + 0.065 = 1.065
\][/tex]
5. Raise the result to the power of 13:
[tex]\[
1.065^{13}
\][/tex]
6. Multiply the principal amount by the result obtained in the previous step:
[tex]\[
A = 28000 \times 1.065^{13}
\][/tex]
7. After performing the calculation, you get:
[tex]\[
A \approx 28000 \times 2.267487497
\][/tex]
[tex]\[
A \approx 63489.65
\][/tex]
8. Round the final amount to the nearest dollar:
[tex]\[
A \approx 63490
\][/tex]
Thus, the amount owed after 13 years, with 6.5% interest compounded annually, is $63,490.
