To determine the amount in the account after 15 years, we will use the compound interest formula:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
where:
- [tex]\( P \)[/tex] is the initial investment (principal)
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal)
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year
- [tex]\( t \)[/tex] is the number of years
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest
Given:
- [tex]\( P = 3000 \)[/tex]
- [tex]\( r = 0.054 \)[/tex] (since 5.4% as a decimal is 0.054)
- [tex]\( n = 2 \)[/tex] (since the interest is compounded semiannually)
- [tex]\( t = 15 \)[/tex]
First, we substitute the given values into the formula:
[tex]\[
A = 3000 \left(1 + \frac{0.054}{2}\right)^{2 \cdot 15}
\][/tex]
Next, we simplify the expression inside the parentheses:
[tex]\[
1 + \frac{0.054}{2} = 1 + 0.027 = 1.027
\][/tex]
Then, we calculate the exponent:
[tex]\[
2 \cdot 15 = 30
\][/tex]
Now, we have:
[tex]\[
A = 3000 \left(1.027\right)^{30}
\][/tex]
Finally, we compute the value of [tex]\(\left(1.027\right)^{30}\)[/tex] and multiply by 3000 to find the amount [tex]\( A \)[/tex].
[tex]\[
A \approx 3000 \times 2.22389 \approx 6671.67
\][/tex]
Therefore, the amount in the account after 15 years, rounded to the nearest hundredths place, is:
[tex]\[
\$ 6671.67
\][/tex]
The correct answer is:
[tex]\[
\boxed{\$ 6671.67}
\][/tex]
