To determine the future value of an investment with compound interest, we use the compound interest formula:
[tex]\[ A(t) = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( P \)[/tex] is the initial principal (the initial amount of money invested),
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal),
- [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 or borrowed for,
- [tex]\( A(t) \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
For this specific problem, we are given:
- [tex]\( P = 200 \)[/tex] dollars,
- [tex]\( r = 7\% = 0.07 \)[/tex] (as a decimal),
- [tex]\( n = 1 \)[/tex] (since the interest is compounded annually),
- [tex]\( t = 5 \)[/tex] years.
Substitute these values into the compound interest formula:
[tex]\[ A(t) = 200 \left(1 + \frac{0.07}{1}\right)^{1 \cdot 5} \][/tex]
[tex]\[ A(t) = 200 \left(1 + 0.07\right)^5 \][/tex]
[tex]\[ A(t) = 200 \left(1.07\right)^5 \][/tex]
Now, we calculate [tex]\( (1.07)^5 \)[/tex]:
[tex]\[ (1.07)^5 \approx 1.402552 \][/tex]
Next, multiply this result by the principal [tex]\( P \)[/tex]:
[tex]\[ A(t) = 200 \times 1.402552 \approx 280.510346 \][/tex]
So, the amount of money after 5 years is approximately [tex]\( \$ 280.510346 \)[/tex].
Rounding this to the nearest cent:
[tex]\[ \$ 280.510346 \approx \$ 280.51 \][/tex]
Therefore, the amount of the investment after 5 years is:
[tex]\[ \boxed{280.51} \][/tex]
Thus, the correct answer is:
[tex]\[ \text{B. } \$ 280.51 \][/tex]
