To find out how much the investment will be worth after 14 years, given that it is compounded monthly, we can use the compound interest formula:
[tex]\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( P \)[/tex] is the principal amount (initial investment)
- [tex]\( r \)[/tex] is the annual interest rate
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year
- [tex]\( t \)[/tex] is the number of years the money is invested for
Given:
- [tex]\( P = 240 \)[/tex] dollars
- [tex]\( r = 9\% = 0.09 \)[/tex] (as a decimal)
- [tex]\( n = 12 \)[/tex] (since the interest is compounded monthly)
- [tex]\( t = 14 \)[/tex] years
Let's plug these values into the formula step-by-step:
1. Calculate the monthly interest rate: [tex]\(\frac{r}{n} = \frac{0.09}{12}\)[/tex]
2. Add the monthly interest rate to 1: [tex]\( 1 + \frac{r}{n} = 1 + \frac{0.09}{12} \)[/tex]
3. Calculate the total number of compounding periods: [tex]\( nt = 12 \times 14 \)[/tex]
4. Raise the result from step 2 to the power of the total number of compounding periods: [tex]\(\left(1 + \frac{0.09}{12}\right)^{12 \times 14}\)[/tex]
5. Multiply the principal amount ([tex]\( P \)[/tex]) by the result from step 4 to find the future value of the investment: [tex]\( A = 240 \times \left(1 + \frac{0.09}{12}\right)^{12 \times 14} \)[/tex]
After performing these calculations, we find:
[tex]\[ A \approx 842.13 \][/tex]
Thus, after 14 years, the investment will be worth approximately \[tex]$842.13.
Therefore, the correct answer is:
\[ \$[/tex] 842.13 \]
