smc

By the Angle Bisector Theorem, we know that $\frac{AB}{BC}=\frac{3}{8}$. If we use the lowest possible integer values for $AB$ and $BC$ (the lengths of $AD$ and $DC$, respectively), then $AB+BC=AD+DC=AC$, contradicting the Triangle Inequality. If we use the next lowest values ($AB=6$ and $BC=16$), the Triangle Inequality is satisfied. Therefore, our answer is $6+16+3+8=\boxed{33}$, or choice $\text{(B)}$.