Answer: 20

Small Hint:

Find the altitudes from $C$ and $D$ to the common edge $AB$

Big Hint:

Express the tetrahedron’s volume using the common edge, the two altitudes, and the sine of the dihedral angle

Solution:

Let $h_C$ and $h_D$ be the perpendicular distances from $C$ and $D$ to $AB.$ From the two face areas, $$\begin{aligned} h_C&=\frac{2\cdot15}{3}=10,\\ h_D&=\frac{2\cdot12}{3}=8. \end{aligned}$$ The angle between these two perpendicular directions is the $30^\circ$ dihedral angle. Hence the scalar triple product gives $$\begin{aligned} V&=\frac16(AB)h_Ch_D\sin30^\circ\\ &=\frac16\cdot3\cdot10\cdot8\cdot\frac12\\ &=20. \end{aligned}$$