To determine how much \[tex]$200 invested at an annual interest rate of 4%, compounded monthly, would be worth after 8 years, we can use the compound interest formula:
\[
A(t) = P \left(1 + \frac{r}{n} \right)^{nt}
\]
where:
- \( A(t) \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money, which is \$[/tex]200 in this case).
- [tex]\( r \)[/tex] is the annual interest rate (decimal form, so 4% becomes 0.04).
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year (monthly compounding means [tex]\( n = 12 \)[/tex]).
- [tex]\( t \)[/tex] is the number of years the money is invested for (8 years).
Let's break down the calculation step by step:
1. Principal, [tex]\( P \)[/tex]:
[tex]\[ P = 200 \][/tex]
2. Annual interest rate, [tex]\( r \)[/tex]:
[tex]\[ r = 0.04 \][/tex]
3. Compounding frequency, [tex]\( n \)[/tex]:
[tex]\[ n = 12 \][/tex]
4. Time in years, [tex]\( t \)[/tex]:
[tex]\[ t = 8 \][/tex]
Plug these values into the compound interest formula:
[tex]\[
A(t) = 200 \left(1 + \frac{0.04}{12}\right)^{12 \times 8}
\][/tex]
Now let's simplify inside the parentheses first:
[tex]\[
1 + \frac{0.04}{12} = 1 + 0.0033333 = 1.0033333
\][/tex]
Next, raise this to the power of [tex]\( 12 \times 8 \)[/tex]:
[tex]\[
(1.0033333)^{96}
\][/tex]
Calculating this exponent:
[tex]\[
(1.0033333)^{96} \approx 1.376395
\][/tex]
Now multiply this with the principal amount:
[tex]\[
200 \times 1.376395 \approx 275.2790238466248
\][/tex]
Finally, we round this amount to the nearest cent:
[tex]\[
\approx 275.28
\][/tex]
Thus, the value of the investment after 8 years is:
[tex]\[\$ 275.28\][/tex]
So the correct answer is (B) [tex]$\$[/tex] 275.28$.
