To find the future value of \[tex]$4000 earning 12% interest, compounded monthly, for 6 years, we'll use the compound interest formula. The compound interest formula is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the future value of the investment/loan, including interest.
- \( P \) is the principal investment amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed for.
Given the problem, we have:
- \( P = 4000 \) dollars
- \( r = 0.12 \) (12% annual interest rate)
- \( n = 12 \) (compounded monthly)
- \( t = 6 \) years
Plugging these values into the formula, we get:
\[
A = 4000 \left(1 + \frac{0.12}{12}\right)^{12 \cdot 6}
\]
First, calculate the monthly interest rate:
\[
\frac{0.12}{12} = 0.01
\]
Next, add 1 to the monthly interest rate:
\[
1 + 0.01 = 1.01
\]
Then, raise this amount to the power of the total number of compounding periods (months in this case):
\[
1.01^{12 \cdot 6} = 1.01^{72}
\]
Using a calculator for the exponentiation:
\[
1.01^{72} \approx 2.04767
\]
Finally, multiply this result by the principal:
\[
4000 \times 2.04767 \approx 8188.4
\]
Therefore, the future value of \$[/tex]4000 earning 12% interest, compounded monthly for 6 years, is approximately \$8188.40 rounded to two decimal places.
