3. In quadrilateral $A B C D$, $B C=8, C D=1$, $\angle A B C=30^{\circ}, \angle B C D=60^{\circ}$. If the area of quadrilateral $A B C D$ is $\frac{13 \sqrt{3}}{2}$, then the value of $A B$ is ( ).
(A) $\sqrt{3}$
(B) $2 \sqrt{3}$
(C) $3 \sqrt{3}$
(D) $4 \sqrt{3}$
3. (C).

As shown in Figure 3, since $\angle A B C$ and $\angle B C D$ are complementary, extending $B A$ and $C D$ to intersect at $E$, we have $\angle B E C=90^{\circ}$, and
$$
\begin{array}{l}
C E=\frac{1}{2} B C=4, \\
B E=\frac{\sqrt{3}}{2} B C=4 \sqrt{3} .
\end{array}
$$

We have $S_{\triangle A D E}=S_{\triangle B C E}-S_{\text {quadrilateral } A B C D}$
$$
\begin{array}{l}
=\frac{1}{2} C E \cdot B E-S_{\text {quadrilateral } A B C D} \\
=8 \sqrt{3}-\frac{13 \sqrt{3}}{2} \\
=\frac{3 \sqrt{3}}{2} .
\end{array}
$$

Also, $D E=C E-C D=3$,
$$
A E=\frac{2 S_{\triangle D E}}{D E}=\sqrt{3} \text {, }
$$

Thus, $A B=B E-A E=4 \sqrt{3}-\sqrt{3}=3 \sqrt{3}$.