To determine how much Claire will owe after ten years if she does not make any payments, we can use the formula for compound interest:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for in years.
Given the problem:
- [tex]\( P = \$ 3,000 \)[/tex] (the principal amount)
- [tex]\( r = 0.15 \)[/tex] (annual interest rate of 15% expressed as a decimal)
- [tex]\( n = 1 \)[/tex] (interest compounds once per year)
- [tex]\( t = 10 \)[/tex] (number of years)
Substitute these values into the formula:
[tex]\[ A = 3000 \left(1 + \frac{0.15}{1}\right)^{1 \cdot 10} \][/tex]
[tex]\[ A = 3000 \left(1 + 0.15\right)^{10} \][/tex]
[tex]\[ A = 3000 \left(1.15\right)^{10} \][/tex]
Using this formula, Claire will owe \[tex]$12,136.67 after ten years.
So, the correct answer is:
\[ \$[/tex] 12,136.67 \]
