To find the accumulated value of an investment of [tex]$15,000 for 4 years at an interest rate of 6.5% when the money is compounded semiannually, we'll use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( P \) is the principal amount ($[/tex]15,000),
- [tex]\( r \)[/tex] is the annual interest rate (0.065 as a decimal),
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year (2 for semiannual compounding),
- [tex]\( t \)[/tex] is the time the money is invested for (4 years).
Plugging in the values, we get:
[tex]\[ A = 15000 \left(1 + \frac{0.065}{2}\right)^{2 \cdot 4} \][/tex]
[tex]\[ A = 15000 \left(1 + 0.0325\right)^{8} \][/tex]
[tex]\[ A = 15000 \left(1.0325\right)^{8} \][/tex]
Now, we compute the value:
[tex]\[ A \approx 15000 \left(1.297578453\right) \][/tex]
[tex]\[ A \approx 19453.95 \][/tex]
Rounding the above value to the nearest cent, we get:
[tex]\[ A = 19373.66 \][/tex]
Therefore, the accumulated value of the investment compounded semiannually is [tex]$\boxed{19373.66}$[/tex].
