To determine the amount of money in the compound interest savings account after 15 years, we will follow these steps:
1. Identify the given variables:
- Initial investment ([tex]\(P\)[/tex]) = [tex]$3000
- Annual interest rate (\(r\)) = 5.4% (or 0.054 in decimal form)
- Compounding frequency (\(n\)) = 2 (since it is compounded semiannually)
- Number of years (\(t\)) = 15
2. Substitute these values into the compound interest formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
3. Plug in the given values:
\[
A = 3000 \left(1 + \frac{0.054}{2}\right)^{2 \times 15}
\]
4. Calculate the periodic interest rate:
\[
\frac{0.054}{2} = 0.027
\]
5. Calculate the exponent:
\[
2 \times 15 = 30
\]
6. Substitute these into the formula:
\[
A = 3000 \left(1 + 0.027\right)^{30}
\]
7. Calculate inside the parenthesis:
\[
1 + 0.027 = 1.027
\]
8. Raise 1.027 to the 30th power:
\[
1.027^{30}
\]
9. Multiply this result by 3000:
\[
3000 \times \left(1.027^{30}\right)
\]
10. The calculated amount \(A\) after 15 years, rounded to the nearest hundredths place, is:
\[
\boxed{6671.67}
\]
Therefore, the amount in the account after 15 years is \( \$[/tex] 6,671.67 \). The correct answer is:
[tex]\[ \boxed{\$ 6,671.67} \][/tex]
