To find how much an investment of \[tex]$240 will be worth after 14 years with an annual interest rate of \( 9\% \) compounded monthly, we will use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( P \) is the principal amount (initial investment), which is \$[/tex]240.
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal), which is [tex]\( 0.09 \)[/tex].
- [tex]\( n \)[/tex] is the number of times interest is compounded per year, which is 12 (monthly compounding).
- [tex]\( t \)[/tex] is the time the money is invested for in years, which is 14.
Let's break it down step-by-step:
1. Convert the annual interest rate to a decimal:
[tex]\[ r = 0.09 \][/tex]
2. Identify the number of compounding periods per year:
[tex]\[ n = 12 \][/tex]
3. Identify the number of years the money is invested:
[tex]\[ t = 14 \][/tex]
4. Substitute the values into the compound interest formula:
[tex]\[ A = 240 \left(1 + \frac{0.09}{12}\right)^{12 \times 14} \][/tex]
5. Calculate the monthly interest rate:
[tex]\[ \frac{0.09}{12} = 0.0075 \][/tex]
6. Calculate the total number of compounding periods:
[tex]\[ 12 \times 14 = 168 \][/tex]
7. Calculate the base of the exponent:
[tex]\[ 1 + 0.0075 = 1.0075 \][/tex]
8. Raise the base to the power of the total number of compounding periods:
[tex]\[ (1.0075)^{168} \approx 3.508885595 \][/tex]
9. Multiply this result by the principal amount:
[tex]\[ A = 240 \times 3.508885595 \approx 842.1325429162007 \][/tex]
Therefore, the value of the investment after 14 years is approximately \[tex]$842.13.
Among the given choices:
- \$[/tex]68.39
- \[tex]$704.28
- \$[/tex]842.13
- \[tex]$846.10
The closest value is \$[/tex]842.13.
So the correct answer is:
[tex]\[ \boxed{842.13} \][/tex]
