Sure, let's break down the problem step by step using the compound interest formula given:
[tex]\[ A=P\left(1+\frac{r}{n}\right)^{n t} \][/tex]
Here:
- [tex]\( P \)[/tex] is the principal amount (initial amount of the loan),
- [tex]\( r \)[/tex] is the annual interest rate,
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year,
- [tex]\( t \)[/tex] is the number of years the money is borrowed for,
- [tex]\( A \)[/tex] is the final amount that needs to be repaid.
Given the details for Loan Option B:
- Principal, [tex]\( P = \$9,000 \)[/tex]
- Annual interest rate, [tex]\( r = 8\% = 0.08 \)[/tex]
- Compounded annually hence, [tex]\( n = 1 \)[/tex]
- Time, [tex]\( t = 3 \)[/tex] years
Substituting these values into the compound interest formula:
[tex]\[ A = 9000 \left( 1 + \frac{0.08}{1} \right)^{1 \cdot 3} \][/tex]
Simplifying inside the parentheses:
[tex]\[ A = 9000 \left( 1 + 0.08 \right)^3 \][/tex]
[tex]\[ A = 9000 \left( 1.08 \right)^3 \][/tex]
[tex]\[ A = 9000 \times 1.259712 \][/tex]
[tex]\[ A \approx 11337.408 \][/tex]
Therefore, the total amount that Tom would repay after 3 years is approximately:
[tex]\[ \boxed{11337.41} \][/tex]
Among the given options, [tex]\( \$ 11,337 \)[/tex] is the closest and correct amount.
