To determine the future value of an investment that is compounded continuously, we can use the formula for continuous compounding:
[tex]\[ A = P e^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money, which is [tex]$25,000 in this case).
- \( r \) is the annual interest rate (expressed as a decimal, so 2.65% becomes 0.0265).
- \( t \) is the time in years (since it is not specified in your question, we will assume it to be 5 years).
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
Given these parameters:
- \( P = 25,000 \)
- \( r = 0.0265 \)
- \( t = 5 \)
Substituting these into the formula:
\[ A = 25,000 \times e^{0.0265 \times 5} \]
First, we calculate the exponent:
\[ 0.0265 \times 5 = 0.1325 \]
Next, we evaluate \( e^{0.1325} \):
\[ e^{0.1325} \approx 1.141898 \]
Now we multiply this by the principal amount:
\[ A = 25,000 \times 1.141898 \]
\[ A \approx 28,541.98 \]
Therefore, the future value of the investment after 5 years, rounded to the nearest penny, is $[/tex]28,541.98.
