To determine how much Alisha will have paid at the end of her five-year loan term, we need to apply the formula for compound interest. This formula allows us to account for interest that is compounded annually. The standard formula is:
[tex]\[ A = P (1 + r)^t \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount (the initial amount of the loan)
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal)
- [tex]\( t \)[/tex] is the number of years the money is invested or borrowed
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
Given the values:
- Principal [tex]\( P = \$15,000 \)[/tex]
- Annual interest rate [tex]\( r = 0.06 \)[/tex] (6 percent)
- Time period [tex]\( t = 5 \)[/tex] years
We'll substitute these values into the formula:
[tex]\[ A = 15000 (1 + 0.06)^5 \][/tex]
Calculating step-by-step:
1. First, add [tex]\( 1 + 0.06 = 1.06 \)[/tex].
2. Then, raise [tex]\( 1.06 \)[/tex] to the power of 5: [tex]\( 1.06^5 \)[/tex].
3. Finally, multiply the result by the principal: [tex]\( 15000 \times 1.3382255776 \approx 20073.383664000005 \)[/tex].
Hence, the total amount Alisha will have to pay at the end of the five-year loan term is:
[tex]\[ \approx \$ 20,073.38 \][/tex]
From the given options, none of them exactly matches this value precisely. However, the closest option is:
C. \$ 20,073.50
