11.7. The area of the surface of a spherical segment is $S$ (the spherical part of its surface is meant). What is the greatest possible volume of such a segment?
11.7. Let $V$ be the volume of a spherical segment, $R$ be the radius of the sphere. Since $S=2 \pi R h$ (problem 4.24) and $V=\pi h^{2}(3 R-$ - $h$ ) $/ 3$ (problem 4.27), then $V=S h / 2-\pi h^{3} / 3$. Therefore, the derivative of $V$ with respect to $h$ is $S / 2-\pi h^{2}$. The maximum volume will be at $h=$ $=\sqrt{S / 2 \pi} ;$ it equals $S \sqrt{S / 18 \pi}$.