To determine the amount of money that a [tex]$100 investment at 6% annual interest compounded monthly would be worth after 20 years, we can use the formula for compound interest:
\[ A(t) = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( P \) is the principal amount (initial investment), which is $[/tex]100.
- [tex]\( r \)[/tex] is the annual interest rate, which is 0.06 (6%).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year. Since the interest is compounded monthly, [tex]\( n = 12 \)[/tex].
- [tex]\( t \)[/tex] is the number of years the money is invested, which is 20.
Let's break down the calculation step-by-step:
1. Identify the given values:
[tex]\[ P = 100 \][/tex]
[tex]\[ r = 0.06 \][/tex]
[tex]\[ n = 12 \][/tex]
[tex]\[ t = 20 \][/tex]
2. Plug the values into the compound interest formula:
[tex]\[ A(t) = 100 \left(1 + \frac{0.06}{12}\right)^{12 \cdot 20} \][/tex]
3. Simplify the fraction inside the parentheses:
[tex]\[ A(t) = 100 \left(1 + 0.005\right)^{240} \][/tex]
4. Calculate the expression inside the parentheses:
[tex]\[ A(t) = 100 \left(1.005\right)^{240} \][/tex]
5. Evaluate the power:
Evaluating [tex]\( (1.005)^{240} \approx 3.310204 \)[/tex]
6. Multiply this result by the principal amount:
[tex]\[ A(t) = 100 \times 3.310204 \][/tex]
[tex]\[ A(t) = 331.02 \][/tex]
After rounding to the nearest cent, the amount of money accumulated would be [tex]\(\$331.02\)[/tex].
Based on the calculations, the correct answer is:
[tex]\[ \boxed{331.02} \][/tex]
Thus, the correct choice is:
[tex]\[ \boxed{\text{A. } \$ 331.02} \][/tex]
