Let's analyze the problem step-by-step to find the amount of money that will be in the account after 15 years through compound interest.
### Step-by-Step Solution:
1. Identify the given values:
- Initial investment ([tex]\(P\)[/tex]): [tex]$3000
- Annual interest rate (\(r\)): 5.4%, which can be expressed as 0.054 when converted to decimal form.
- Number of times the interest is compounded per year (\(n\)): 2 (since it is compounded semiannually)
- Time (\(t\)): 15 years
2. Use the compound interest formula:
The formula to calculate the amount of money accumulated, \(A\), using compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
3. Substitute the given values into the formula:
\[
A = 3000 \left(1 + \frac{0.054}{2}\right)^{2 \times 15}
\]
4. Simplify the expression inside the parentheses:
\[
\left(1 + \frac{0.054}{2}\right) = \left(1 + 0.027\right) = 1.027
\]
5. Calculate the exponent \(2t\):
\[
2 \times 15 = 30
\]
6. Raise the base to the power of 30:
\[
(1.027)^{30}
\]
7. Multiply the initial investment by this result:
\[
A = 3000 \times (1.027)^{30}
\]
8. Round the final amount to the nearest hundredth:
\[
A \approx 6671.67
\]
Therefore, the amount in the account after 15 years, rounded to the nearest hundredth, is:
\[
\$[/tex]6671.67
\]
Given the multiple-choice answers:
- [tex]$\$[/tex]3164.19[tex]$
- \$[/tex]6671.67[tex]$
- $[/tex]\[tex]$4473.81$[/tex]
- [tex]$\$[/tex]14532.47[tex]$
The correct answer is \(\$[/tex]6671.67\).
