To calculate the future value of an investment using the formula for compound interest, we follow a series of steps. Here's a detailed, step-by-step solution to find the future value when [tex]\( P = \$22,000 \)[/tex], [tex]\( r = 5\% \)[/tex] compounded monthly, and [tex]\( t = 9 \)[/tex] years:
1. Identify the known values:
- Principal amount ([tex]\( P \)[/tex]): \[tex]$22,000
- Annual interest rate (\( r \)): 5% or 0.05 in decimal form
- Number of times interest is compounded per year (\( n \)): 12 (since the interest is compounded monthly)
- Number of years the money is invested (\( t \)): 9 years
2. Compound interest formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Here, \( A \) is the future value of the investment.
3. Substitute the known values into the formula:
\[
A = 22000 \left(1 + \frac{0.05}{12}\right)^{12 \times 9}
\]
4. Perform the calculations inside the parentheses first:
\[
\frac{0.05}{12} = 0.00416666666667
\]
Add 1 to this result:
\[
1 + 0.00416666666667 = 1.00416666666667
\]
5. Calculate the exponent \( 12 \times 9 \):
\[
12 \times 9 = 108
\]
6. Raise the base \( 1.00416666666667 \) to the power of 108:
\[
(1.00416666666667)^{108} \approx 1.566282731
\]
7. Multiply this result by the principal amount \(\$[/tex]22,000\):
[tex]\[
22000 \times 1.566282731 \approx 34470.628082
\][/tex]
8. Round the result to two decimal places:
[tex]\[
34470.628082 \approx 34470.63
\][/tex]
Therefore, the future value of the investment is approximately \$34,470.63.
