Problem 3. The area of the base of a right triangular prism is $4 \mathrm{~cm}^{2}$, and the areas of the lateral faces are $9 \mathrm{~cm}^{2}, 10 \mathrm{~cm}^{2}$, and $17 \mathrm{~cm}^{2}$. Calculate the volume of the prism.
Solution. Let $H$ be the height of the prism. Then $a=\frac{9}{H}, b=\frac{10}{H}, c=\frac{17}{H}$, where $a, b, c$ are the lengths of the edges of the base. From $B=4 \, \text{cm}^2$, by applying Heron's formula, we get $4=\frac{36}{H^2}$, from which it follows that $H=3 \, \text{cm}, V=B H=12 \, \text{cm}^3$.