To determine how much \[tex]$200 invested at 5% interest compounded monthly would be worth after 9 years, we use the formula for compound interest:
\[
A(t) = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- \(P\) is the principal amount (the initial investment),
- \(r\) is the annual interest rate (as a decimal),
- \(n\) is the number of times interest is compounded per year,
- \(t\) is the number of years the money is invested,
- \(A(t)\) is the amount of money accumulated after \(t\) years, including interest.
Let's break down the values given:
- \(P = \$[/tex]200\)
- [tex]\(r = 0.05\)[/tex] (5% annual interest rate)
- [tex]\(n = 12\)[/tex] (since interest is compounded monthly)
- [tex]\(t = 9\)[/tex] years
Now, substitute these values into the formula:
[tex]\[
A(9) = 200 \left(1 + \frac{0.05}{12}\right)^{12 \times 9}
\][/tex]
First, calculate the monthly interest rate:
[tex]\[
\frac{0.05}{12} \approx 0.004167
\][/tex]
Next, compute the exponent:
[tex]\[
12 \times 9 = 108
\][/tex]
Now calculate the expression inside the parentheses:
[tex]\[
1 + 0.004167 \approx 1.004167
\][/tex]
Raise this to the power of 108:
[tex]\[
(1.004167)^{108} \approx 1.566298
\][/tex]
Finally, multiply by the initial investment, [tex]\(P\)[/tex]:
[tex]\[
A(9) = 200 \times 1.566298 = 313.3696
\][/tex]
Rounding this to the nearest cent, we get:
[tex]\[
\$313.37
\][/tex]
Therefore, the correct answer is:
A. [tex]\(\$313.37\)[/tex]
