The area of the base of a right triangular prism is 4, the areas of the lateral faces are 9, 10, and 17. Find the volume of the prism.

#
Let $V$ be the volume of the prism; $h$ be its height; $a, b$, and $c$ be the sides of the base of the prism; $p$ be the semi-perimeter of the base, and $S$ be the area of the base. According to the problem,

$$
a h=9, b h=10, c h=17
$$

from which

$$
a=\frac{9}{h}, b=\frac{19}{h}, c=\frac{17}{h} .
$$

Then

$$
p=\frac{1}{2}\left(\frac{9}{h}+\frac{19}{h}+\frac{17}{h}\right)=\frac{18}{h}, S=\sqrt{p(p-a)(p-b)(p-c)}=\sqrt{\frac{18}{h} \cdot \frac{9}{h} \cdot \frac{8}{h} \cdot \frac{1}{h}}=\frac{3 \frac{1}{h}}{h}=4
$$

from which we find that $h=3$. Therefore,

$$
V=S h=4 \cdot 3=12 .
$$

## Answer

12.00