To calculate the future value of an investment with the given parameters, we need to use the formula for compound interest. The formula to calculate the future value [tex]\( A \)[/tex] is given by:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{n \cdot t} \][/tex]
Here's a detailed step-by-step solution using the given values:
1. Identify the given values:
- Principal amount ([tex]\( P \)[/tex]) = \[tex]$21,000
- Annual interest rate (\( r \)) = 7\% or 0.07 (as a decimal)
- Number of times the interest is compounded per year (\( n \)) = 12 (monthly)
- Number of years (\( t \)) = 5
2. Plug the values into the formula:
\[ A = 21,000 \left(1 + \frac{0.07}{12}\right)^{12 \cdot 5} \]
3. Calculate the periodic interest rate (\( \frac{r}{n} \)):
\[ \frac{0.07}{12} \approx 0.0058333333 \]
4. Add the periodic interest rate to 1:
\[ 1 + 0.0058333333 \approx 1.0058333333 \]
5. Calculate the exponent (\( n \cdot t \)):
\[ 12 \cdot 5 = 60 \]
6. Raise the base to the power of the exponent:
\[ (1.0058333333)^{60} \approx 1.41758 \]
7. Multiply the principal amount by the result from the previous step:
\[ 21,000 \cdot 1.41758 \approx 29770.1358 \]
8. Round the final answer to two decimal places:
\[ 29770.1358 \approx 29770.13 \]
Therefore, the future value of the investment is \$[/tex]29,770.13.
