To solve the question of finding the value of an ordinary annuity given the semiannual payments, the interest rate, and the number of years, we can follow these steps:
1. Identify the given parameters:
- Amount of each payment: \[tex]$11,800
- Payment frequency: Semiannually
- Total number of years: 8
- Annual interest rate: 7%
2. Determine the total number of payments:
Since payments are made semiannually (twice a year), and there are 8 years:
\[
\text{Total payments} = 8 \text{ years} \times 2 \text{ payments per year} = 16 \text{ payments}
\]
3. Convert the annual interest rate to the interest rate per period:
Given the annual interest rate is 7% and payments are made semiannually, we need to divide the annual rate by 2:
\[
\text{Periodic interest rate} = \frac{7\%}{2} = 3.5\% = 0.035
\]
4. Apply the formula for the future value of an ordinary annuity:
The formula to calculate the future value of an ordinary annuity is:
\[
\text{Annuity value} = \text{Payment} \times \left( \frac{(1 + r)^n - 1}{r} \right)
\]
where
\[
r = \text{Periodic interest rate} = 0.035 \quad \text{and} \quad n = \text{Total number of payments} = 16
\]
5. Plug in the values into the formula:
\[
\text{Annuity value} = 11800 \times \left( \frac{(1 + 0.035)^{16} - 1}{0.035} \right)
\]
6. Compute the annuity value:
\[
\text{Annuity value} = 11800 \times \left( \frac{(1.035)^{16} - 1}{0.035} \right)
\]
Evaluating this expression step-by-step yields the future value of the annuity.
Based on these computations, the final result for the value of the annuity is:
\[
\text{Annuity value} = 247458.15
\]
Thus, the value of the annuity, rounded to the nearest cent, is \$[/tex]247,458.15.
