To solve this problem, we'll use the compound interest formula to determine the amount owed after 10 years. The compound interest formula is given by:
[tex]\[ A = P(1 + \frac{r}{n})^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount owed after time [tex]\( t \)[/tex],
- [tex]\( P \)[/tex] is the principal amount (initial amount of money),
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form),
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year,
- [tex]\( t \)[/tex] is the time in years.
Given the provided data:
- [tex]\( P = 35,000 \)[/tex] dollars,
- [tex]\( r = 5.5\% = 0.055 \)[/tex],
- [tex]\( t = 10 \)[/tex] years,
- [tex]\( n = 1 \)[/tex] (since the interest is compounded annually).
Plug these values into the formula to find the amount owed [tex]\( A \)[/tex]:
[tex]\[ A = 35,000 \left(1 + \frac{0.055}{1}\right)^{1 \times 10} \][/tex]
Simplify the equation step-by-step:
1. Calculate the interest rate per compounding period:
[tex]\[ 1 + \frac{0.055}{1} = 1 + 0.055 = 1.055 \][/tex]
2. Raise this value to the power of the total number of compounding periods:
[tex]\[ (1.055)^{10} \][/tex]
3. Multiply the principal amount by this value:
[tex]\[ 35,000 \times (1.055)^{10} \][/tex]
Through the calculation, you will find that the amount owed after 10 years is approximately:
[tex]\[ A \approx 59,785 \][/tex]
Thus, the amount owed after 10 years is $59,785, rounded to the nearest dollar.
