To determine how much Alisha will have paid at the end of the five-year loan term, we will use the compound interest formula:
[tex]\[
\text{total amount} = P(1 + i)^t
\][/tex]
Where:
- [tex]\(P\)[/tex] is the principal amount (the initial loan amount),
- [tex]\(i\)[/tex] is the annual interest rate (expressed as a decimal),
- and [tex]\(t\)[/tex] is the duration of the loan in years.
Given:
- [tex]\(P = \$15,000\)[/tex]
- [tex]\(i = 0.06\)[/tex] (6 percent interest rate)
- [tex]\(t = 5\)[/tex] years
Let's substitute these values into the compound interest formula:
[tex]\[
\text{total amount} = 15000 \times (1 + 0.06)^5
\][/tex]
First, we calculate [tex]\(1 + 0.06\)[/tex]:
[tex]\[
1 + 0.06 = 1.06
\][/tex]
Next, we need to raise 1.06 to the 5th power:
[tex]\[
1.06^5 \approx 1.338225
\][/tex]
Then, we multiply the principal amount by this result:
[tex]\[
15000 \times 1.338225 \approx 20073.38
\][/tex]
Therefore, at the end of the five-year loan term, Alisha will have paid approximately \[tex]$20,073.38.
Now, let's find the closest matching answer from the options provided:
A. \$[/tex] 19,500.25
B. \[tex]$ 15,900.50
C. \$[/tex] 20,073.50
The closest answer is:
C. \$ 20,073.50
