This is a compound interest problem. Since the interest is compounded semi-annually (twice per year), we use the compound interest formula:

$$
\(A=P\left(1+\frac{r}{n}\right)^{n t}\)
$$


Where:
- $\(P=\$ 10,000\)$ (Principal)
- $\(r=0.0396\)$ (Annual interest rate, or $3.96 %$ )
- $n=2$ (Number of times compounded per year)
- $t=2$ (Number of years)
1. Calculate the values needed for the formula:
- Interest rate per period ( $\(\frac{r}{n}\)$ ):

$$
\(\frac{0.0396}{2}=0.0198\)
$$

- Total number of periods ( $n t$ ):

$$
\(2 \times 2=4\)
$$

2. Substitute the values and calculate the final amount ( $A$ ):

$$
\(\begin{gathered}
A=10000(1+0.0198)^4 \\
A=10000(1.0198)^4
\end{gathered}\)
$$

Calculate the exponent:

$$
\((1.0198)^4 \approx 1.0815827\)
$$


Calculate the final amount:

$$
\(\begin{gathered}
A \approx 10000 \times 1.0815827 \\
A \approx 10815.827
\end{gathered}\)
$$


Rounding to the nearest cent gives: $\$ \(10,815.83\)$.

3. Compare with the options:

The calculated amount is closest to B. $\(\mathbf{\$ 1 0 8 1 5 . 8 3}\)$.