[ Volume of a parallelepiped ]

The base of an oblique prism is a parallelogram with sides 3 and 6 and an acute angle of $45^{\circ}$. The lateral edge of the prism is 4 and is inclined to the base plane at an angle of $30^{\circ}$. Find the volume of the prism.
Let $K$ be the orthogonal projection of vertex $A$ of the oblique prism $A B C D A 1 B 1 C 1 D 1$ onto the base plane $A B C D, A B=3, A D=6, \angle B A D=45^{\circ}, A A 1=B B 1=C C 1=D D 1=4$. According to the problem, $\angle A A 1 K=30^{\circ}$. From the right triangle $A K A 1$, we find that $A K=\frac{1}{2} A A 1=2$, and since $A K$ is the height of the prism $A B C D A 1 B 1 C 1 D 1$, then

$$
\begin{aligned}
V_{A B C D A} 1 B 1 C 1 D 1 & =S_{A B C D} \cdot A K=A B \cdot A D \sin 45^{\circ} \cdot A K= \\
& =3 \cdot 6 \cdot \frac{1}{2} \sqrt{2} \cdot 2=18 \sqrt{2}
\end{aligned}
$$