Answered

Use the compound-interest formula to find the account balance [tex]\( A \)[/tex], where [tex]\( P \)[/tex] is the principal, [tex]\( r \)[/tex] is the interest rate, [tex]\( n \)[/tex] is the number of compounding periods per year, [tex]\( t \)[/tex] is time in years, and [tex]\( A \)[/tex] is the account balance.

Given:
\begin{tabular}{|c|c|c|c|}
\hline
[tex]$P$[/tex] & [tex]$r$[/tex] & Compounded & [tex]$t$[/tex] \\
\hline
\[tex]$11,391 & 6.8\% & Daily & 5 \\
\hline
\end{tabular}

The account balance is approximately \$[/tex] [tex]\(\square\)[/tex].

(Simplify your answer. Do not round until the final answer. Then round to two decimal places as needed.)



Answer :

GinnyAnswer
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]