To determine the amount of money that will be in the account at the end of 9 years given an initial deposit of $9000, an annual interest rate of 6%, compounded semi-annually, we will use the compound interest formula:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{tn}
\][/tex]
Here, \(P\) is the principal amount (initial deposit), \(r\) is the annual interest rate, \(n\) is the number of times the interest is compounded per year, and \(t\) is the number of years.
1. Identify the variables:
[tex]\[
\begin{align*}
P &= 9000 \\
r &= 0.06 \\
n &= 2 \\
t &= 9
\end{align*}
\][/tex]
2. Substitute the variables into the compound interest formula:
[tex]\[
A = 9000 \left(1 + \frac{0.06}{2}\right)^{9 \times 2}
\][/tex]
3. Simplify the expression inside the parentheses:
[tex]\[
A = 9000 \left(1 + 0.03\right)^{18}
\][/tex]
[tex]\[
A = 9000 \left(1.03\right)^{18}
\][/tex]
4. Calculate \((1.03)^{18}\):
The approximate value of \((1.03)^{18}\) is \(1.702433028\).
5. Multiply this value by the principal amount \(9000\):
[tex]\[
A = 9000 \times 1.702433028
\][/tex]
[tex]\[
A \approx 15321.897551159142
\][/tex]
6. Round the result to the nearest cent:
[tex]\[
A \approx \boxed{15321.90}
\][/tex]
Thus, the amount of money in the account after 9 years will be approximately [tex]\(\$15,321.90\)[/tex].
