11.059. The areas of the lateral faces of a right triangular prism are $M, N$ and $P$. The lateral edge of the prism is $l$. Determine the volume of the prism.
Solution.

According to the condition $S_{A^{\prime} B_{B} B A}=M, S_{B_{1} C_{1} B C}=N, S_{A_{1} A C C_{1}}=P, C C_{1}=l$

(Fig. 11.54). Then $B C=\frac{N}{l} ; A B=\frac{M}{l} ; A C=\frac{P}{l}$;

$$
V=S \cdot H=S \cdot l=l \cdot \sqrt{\frac{N+M+P}{2 l} \cdot\left(\frac{N+P-M}{2 l}\right) \cdot\left(\frac{N+M-P}{2 l}\right) \cdot\left(\frac{M+P-N}{2 l}\right)}
$$

(we find $S$ using Heron's formula). Thus,

$$
\begin{aligned}
& V=\frac{1}{4 l} \sqrt{(N+M+P)(M+N-P)(M+P-N)(N+P-M)} \\
& \text { Answer: } \quad \frac{1}{4 l} \sqrt{(N+M+P)(M+N-P)(M+P-N)(N+P-M)}
\end{aligned}
$$