To determine how much [tex]$500 invested at an 8% interest rate compounded continuously would be worth after 3 years, we will use the formula for continuous compounding:
\[
A = P e^{rt}
\]
where:
- \( A \) is the amount of money accumulated after \( t \) years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
- \( r \) is the annual interest rate (decimal).
- \( t \) is the time the money is invested for, in years.
For our values:
- \( P = 500 \) (the principal amount)
- \( r = 0.08 \) (the interest rate as a decimal)
- \( t = 3 \) (time in years)
The formula becomes:
\[
A = 500 \cdot e^{0.08 \cdot 3}
\]
First, we calculate the exponent:
\[
0.08 \times 3 = 0.24
\]
Next, we calculate \( e^{0.24} \). Using the exponential function:
\[
e^{0.24} \approx 1.271249
\]
Now, we multiply this by the principal amount:
\[
A = 500 \cdot 1.271249 \approx 635.624575
\]
Rounding the result to the nearest cent, we get:
\[
A \approx 635.62
\]
After 3 years, the amount of money accumulated is approximately $[/tex]\[tex]$ 635.62$[/tex].
Hence, the correct answer is:
D. [tex]$\$[/tex] 635.61$
