4. The areas of the two bases of a truncated pyramid are $Q$ and $G$, and the height is $H$. Determine the volume of the entire pyramid and the volume of the truncated pyramid.
Solution. Let the height of the truncated pyramid be $h$. Then the height of the whole pyramid is $H+h$. According to the theorem of parallel sections in pyramids, we have:

$$
\frac{Q}{G}=\frac{(H+h)^{2}}{h^{2}}
$$

from which we get $h=H \frac{\sqrt{G}}{\sqrt{Q}-\sqrt{G}}$. According to this, the volume of the whole pyramid is

$$
V_{c}=\frac{1}{3} Q(H+h)=\frac{1}{3} \frac{Q H \sqrt{Q}}{\sqrt{Q}-\sqrt{G}}
$$

and the volume of the truncated pyramid is

$$
V_{\circ}=\frac{1}{3} G \cdot h=\frac{1}{3} G \cdot H \frac{\sqrt{G}}{\sqrt{Q}-\sqrt{G}}
$$

i.e.

$$
V_{c}=\frac{H Q(Q+\sqrt{Q G}}{3(Q-G)} \text { and } V_{0}=\frac{H G(G+\sqrt{Q G})}{3(Q-G)}
$$

## Third Year