To determine how much [tex]$400 would be worth after 16 years when invested at an annual interest rate of 3% compounded annually, we can use the compound interest formula:
\[
A(t) = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- \( P \) is the principal amount (initial investment)
- \( r \) is the annual interest rate (in decimal form)
- \( n \) is the number of times the interest is compounded per year
- \( t \) is the time the money is invested for in years
Given:
- \( P = 400 \)
- \( r = 0.03 \) (3% as a decimal)
- \( n = 1 \) (compounded annually)
- \( t = 16 \)
Let’s plug these values into the formula and solve for \( A(t) \):
\[
A(t) = 400 \left(1 + \frac{0.03}{1}\right)^{1 \times 16}
\]
Simplify inside the parentheses:
\[
A(t) = 400 \left(1 + 0.03\right)^{16}
\]
\[
A(t) = 400 \left(1.03\right)^{16}
\]
Now calculate \( 1.03^{16} \) which is approximately:
\[
1.03^{16} \approx 1.604938279
\]
Then multiply this approximate result by 400:
\[
A(t) = 400 \times 1.604938279 \approx 641.88
\]
So, the final amount after 16 years, rounded to the nearest cent, would be approximately:
\[
A(t) \approx \$[/tex]641.88
\]
Thus, the correct answer is:
[tex]\[
\boxed{641.88}
\][/tex]
