smc

Let $D$ be the foot of the altitude to $\overline{B}C$. Then $BD=\sqrt{13^2-12^2}=5$ and $DC=\sqrt{15^2-12^2}=9$. Thus $BC=BD+BC=5+9=14$ or assume that the triangle is obtuse at angle $B$ then $BC=DC-BD=9-5=4$. The sum of the two possible values is $14+4=\boxed{18}$. The answer is $(\mathrm{D})$.