To solve this problem, you need to calculate the amount of money accumulated after a certain time when the interest is compounded continuously. The appropriate formula for continuous compounding is:
[tex]\[ A = P e^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after time [tex]\( t \)[/tex],
- [tex]\( P \)[/tex] is the initial principal balance (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal),
- [tex]\( t \)[/tex] is the time in years,
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately equal to 2.71828).
Given the values:
- [tex]\( P = 7000 \)[/tex] (the initial investment),
- [tex]\( r = 0.04 \)[/tex] (the interest rate, 4% expressed as a decimal),
- [tex]\( t = 8 \)[/tex] years,
we substitute these values into the formula:
[tex]\[ A = 7000 e^{0.04 \times 8} \][/tex]
Using this formula, the amount [tex]\( A \)[/tex] after 8 years is found to be:
[tex]\[ A \approx 9639.89 \][/tex]
So, for the multiple-choice question:
Choose the correct formula and values:
A. [tex]\( A = Pe^{rt} \)[/tex], with [tex]\( P = 7000 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 8 \)[/tex]
Therefore, the amount after 8 years will be \$ 9639.89.
