11.020. The base of an oblique prism is a parallelogram with sides of 3 dm and 6 dm and an acute angle of $45^{\circ}$. The lateral edge of the prism is 4 dm and is inclined to the base plane at an angle of $30^{\circ}$. Find the volume of the prism.
## Solution.

Let $A B=3$ (dm), $A D=6$ (dm), $A A_{1}=4$ (dm) (Fig. 11.16); $A_{1} H-$ height. Then $A_{1} H=A A_{1} \sin 30^{\circ}=2$ (dm). The area of the base $S_{\text {base }}=A B \cdot A D \sin 45^{\circ}=9 \sqrt{2}\left(\right.$ dm $\left.{ }^{2}\right)$. Volume

$V=S_{\text {base }} \cdot A_{1} H=9 \sqrt{2} \cdot 2=18 \sqrt{2}\left(\right.$ dm $\left.^{3}\right)$.

Answer: $18 \sqrt{2}$ dm $^{3}$.