3.1. A plane parallel to the base of a quadrilateral pyramid with a volume of 81 cuts off a smaller pyramid with a volume of 3. Find the volume of the pyramid whose four vertices coincide with the vertices of the section, and the fifth vertex lies in the base plane of the larger pyramid.
Answer: 6. Solution. If we denote the first given volume as $V$, and the second as $v$ (in this case $V=81, v=3$), then the larger pyramid and the smaller pyramid are similar with a similarity coefficient of $\sqrt[3]{\frac{V}{v}}$. Therefore, the height $H$ of the larger pyramid relates to the height $h$ of the smaller pyramid as $\sqrt[3]{\frac{V}{v}}$. Since the height of the pyramid whose volume we need to find is $H-h$, then $\frac{H-h}{h}=\sqrt[3]{\frac{V}{v}}-1$. Therefore, the volume of the pyramid we are looking for is $v\left(\sqrt[3]{\frac{V}{v}}-1\right)$.