To determine the amount of money in a compound interest savings account after 15 years, we can follow these steps:
1. Identify the variables in the compound interest formula:
- Initial investment (principal), [tex]\( P \)[/tex]: \[tex]$3000
- Annual interest rate, \( r \): 5.4% or 0.054 in decimal form
- Number of times interest is compounded per year, \( n \): 2 (semiannually)
- Number of years the money is invested, \( t \): 15
2. Use the compound interest formula:
\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]
Here, \( A \) is the amount of money accumulated after \( t \) years, including interest.
3. Plug in the values into the formula:
\[
A = 3000 \left( 1 + \frac{0.054}{2} \right)^{2 \times 15}
\]
4. Calculate the value inside the parenthesis first:
\[
1 + \frac{0.054}{2} = 1 + 0.027 = 1.027
\]
5. Raise this value to the power of \( 2 \times 15 = 30 \):
\[
1.027^{30}
\]
6. Multiply the result by the principal \( P \):
\[
A = 3000 \times 1.027^{30}
\]
7. Perform the calculations to determine \( A \):
- Calculate \( 1.027^{30} \approx 2.22389 \)
- Multiply this by 3000:
\[
3000 \times 2.22389 = 6671.670091129403
\]
8. Round the result to the nearest hundredths place:
\[
6671.670091129403 \approx 6671.67
\]
Therefore, the amount of money in the account after 15 years, rounded to the nearest hundredths place, is $[/tex]\[tex]$ 6,671.67$[/tex].
Among the given options:
- [tex]$3,164.19$[/tex]
- [tex]$\$[/tex] 6,671.67[tex]$
- $[/tex]\[tex]$ 4,473.81$[/tex]
- [tex]$\$[/tex] 14,532.47[tex]$
The correct answer is \$[/tex] 6,671.67.
