1. The base of the prism is a square with side $a$. The length of the side $a$ is twice the length of the height of the prism. The numerical values of the surface area and volume of the prism are equal. Determine the lengths of the edges of this prism.
Solution. The height of the prism is $h=\frac{a}{2}$. The surface area of the prism is $P=2 a^{2}+4 a h=2 a^{2}+2 a^{2}=4 a^{2}$, and the volume is $V=a^{2} h=\frac{a^{3}}{2}$. From the equality of the surface area and the volume, we get $4 a^{2}=\frac{a^{3}}{2}$ and since $a \neq 0$, from the last equation we get $a=8$. Finally, $h=4$.