To solve for the balance after 15 years when an initial principal is invested with compound interest, we'll use the compound interest formula:
[tex]\[ A = P(1 + r)^n \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( n \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( n \)[/tex] is the number of years the money is invested.
Given the values:
- [tex]\( P = \$ 300 \)[/tex]
- [tex]\( r = 8.5\% = 0.085 \)[/tex]
- [tex]\( n = 15 \)[/tex] years
We plug these values into the compound interest formula:
[tex]\[ A = 300 (1 + 0.085)^{15} \][/tex]
First, calculate [tex]\( 1 + r \)[/tex]:
[tex]\[ 1 + 0.085 = 1.085 \][/tex]
Next, raise this base to the power of [tex]\( n \)[/tex]:
[tex]\[ 1.085^{15} \approx 3.399742 \][/tex]
Now, multiply this result by [tex]\( P \)[/tex] to find [tex]\( A \)[/tex]:
[tex]\[ A = 300 \times 3.399742 = 1019.9228636463336 \][/tex]
Therefore, the balance after 15 years is approximately:
[tex]\[ \$ 1019.92 \][/tex]
Among the given options, the correct answer is:
[tex]\[ \$ 1019.92 \][/tex]
