Certainly! Let's solve the problem step-by-step.
We are given the following information:
- Initial principal, [tex]\( P = \$52700 \)[/tex]
- Annual interest rate, [tex]\( r = 3.2\% \)[/tex] (expressed as a decimal [tex]\( r = 0.032 \)[/tex])
- Time, [tex]\( t = 15 \)[/tex] years
We need to find the value of the account after 15 years using the continuously compounded interest formula:
[tex]\[ V = P e^{r t} \][/tex]
where:
- [tex]\( V \)[/tex] is the value of the account after time [tex]\( t \)[/tex].
- [tex]\( P \)[/tex] is the principal amount (initial investment).
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( t \)[/tex] is the time the money is invested for.
Following the steps:
1. Substitute the known values into the formula:
[tex]\[
V = 52700 \cdot e^{0.032 \cdot 15}
\][/tex]
2. Calculate the exponent:
[tex]\[
0.032 \cdot 15 = 0.48
\][/tex]
3. Raise [tex]\( e \)[/tex] to the power of 0.48:
[tex]\[
e^{0.48}
\][/tex]
4. Multiply the result by the principal:
[tex]\[
V = 52700 \cdot e^{0.48}
\][/tex]
Using the value given by the calculation for [tex]\( e^{0.48} \)[/tex]:
[tex]\[ e^{0.48} \approx 1.615540 \][/tex]
Hence,
[tex]\[ V = 52700 \cdot 1.615540 \approx 85167.12099556548 \][/tex]
5. Round to the nearest cent:
[tex]\[
V \approx 85167.12
\][/tex]
Therefore, after 15 years, the amount of money in the investment account, to the nearest cent, is \$85167.12.
