There is an interesting one class of geometric optimization problems, a couple of which are described here.
http://www.twt.mpei.ac.ru/ochkov/Acad-Hat.pdf

Attached and as a picture SMath solution for one such problem.
I offer SMath-fans such optimization problems.
• a cylinder with a hemispherical bottom and a flat round lid;
• cylinder with hemispherical bottom and without cover;
• a cylinder partitioned off by an inclined elliptical partition so that the vertices of the elliptical partition touch the edges of the circles forming the bottom and the lid of such a container;
• cone without cover (with and without vertical baffle);
• cone with lid (with and without vertical baffle);
• a cone covered with a hemisphere (with and without a vertical baffle);
• a truncated right circular cone with a small base diameter half that of the large base;
• a truncated straight circular cone with the area of the small base half as large as the area of the large base;
• the lower part of the cylinder, dissected by a plane so that it passes through the edge of the circle forming the lid at the top, and at the bottom through the diameter of the circle forming the bottom;
• the upper part of the dissected cylinder described above.
Come up with other similar containers with two parameters to optimize them with constraints.
The task can be simplified - reduced to a flat one. A rectangle with a minimum perimeter length and a given area is a square. And what happens if you do not take into account one side of the square (an inverted letter P with an area limited by three sides)? And what will happen if we make a partition for such a “flat container” and turn it into the letter Ш?

They must be solved analytically. If it does not work out, then numerically
Cylinder-Square.sm (268 КиБ) скачан 92 раз(а).