To solve this problem, we need to determine the present value of an investment that will grow to [tex]$338,893 after 26 years with a 6% interest rate compounded semiannually.
The formula for compound interest is given by:
\[ S = P \left(1 + \frac{r}{m}\right)^{mt} \]
where:
- \( S \) is the future value ($[/tex]338,893)
- [tex]\( P \)[/tex] is the present value (the amount to be invested)
- [tex]\( r \)[/tex] is the annual interest rate (0.06 for 6%)
- [tex]\( m \)[/tex] is the number of times interest is compounded per year (2 for semiannually)
- [tex]\( t \)[/tex] is the number of years the money is invested (26)
We need to solve for [tex]\( P \)[/tex]. Rearrange the formula to solve for the present value [tex]\( P \)[/tex]:
[tex]\[ P = \frac{S}{\left(1 + \frac{r}{m}\right)^{mt}} \][/tex]
Now, substitute the known values into the equation:
[tex]\[ P = \frac{338,893}{\left(1 + \frac{0.06}{2}\right)^{2 \times 26}} \][/tex]
Calculate the term inside the parentheses first:
[tex]\[ \left(1 + \frac{0.06}{2}\right) = 1.03 \][/tex]
Next, calculate the exponent:
[tex]\[ 2 \times 26 = 52 \][/tex]
Now, raise the term inside the parentheses to the exponent:
[tex]\[ 1.03^{52} \approx 4.650791682 \][/tex]
Finally, divide the future value by this result to find the present value:
[tex]\[ P = \frac{338,893}{4.650791682} \approx 72,866.33 \][/tex]
Therefore, the correct answer is:
C. [tex]\(\$ 72,866.33\)[/tex]
