To find the account balance using the compound interest formula, we can proceed as follows:
1. Identify the given parameters:
[tex]\[
P = 11391 \quad (\text{Principal})
\][/tex]
[tex]\[
r = 0.068 \quad (\text{Annual interest rate as a decimal})
\][/tex]
[tex]\[
n = 365 \quad (\text{Number of compounding periods per year})
\][/tex]
[tex]\[
t = 5 \quad (\text{Time in years})
\][/tex]
2. Recall the compound interest formula:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
3. Substitute the given values into the formula:
[tex]\[
A = 11391 \left(1 + \frac{0.068}{365}\right)^{365 \times 5}
\][/tex]
4. Simplify the expression inside the parentheses:
[tex]\[
\frac{0.068}{365} \approx 0.000186301
\][/tex]
[tex]\[
1 + 0.000186301 \approx 1.000186301
\][/tex]
5. Calculate the exponent [tex]\( nt \)[/tex]:
[tex]\[
365 \times 5 = 1825
\][/tex]
6. Raise the base inside the parentheses to the power of [tex]\( 1825 \)[/tex]:
[tex]\[
\left(1.000186301\right)^{1825} \approx 1.40498464
\][/tex]
7. Multiply the result by the principal [tex]\( P \)[/tex]:
[tex]\[
A = 11391 \times 1.40498464 \approx 16003.251216333178
\][/tex]
8. Round the result to two decimal places:
[tex]\[
A \approx 16003.25
\][/tex]
So, the account balance after [tex]\( 5 \)[/tex] years is approximately \$16003.25.
[tex]\[
\boxed{16003.25}
\][/tex]
