To solve the problem of finding the future value of an investment with continuous compounding, we use the continuous compound interest formula:
[tex]\[ A = Pe^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
Given:
- [tex]\( P = 535 \)[/tex] dollars
- [tex]\( r = 0.07 \)[/tex] (7% annual interest rate)
- [tex]\( t = 15 \)[/tex] years
Substitute the given values into the formula:
[tex]\[ A = 535 \times e^{(0.07 \times 15)} \][/tex]
We need to compute the exponent first:
[tex]\[ 0.07 \times 15 = 1.05 \][/tex]
Now, find [tex]\( e \)[/tex] raised to that power:
[tex]\[ e^{1.05} \approx 2.85765 \][/tex]
Next, multiply this result by the principal:
[tex]\[ A = 535 \times 2.85765 \approx 1528.84 \][/tex]
Thus, the investment after 15 years, with continuous compounding at an annual interest rate of 7%, will be approximately worth \[tex]$1528.84.
Therefore, given the choices, the correct answer is:
\[ \$[/tex] 1,528.84 \]
