Given a piece of of circular plate with negligible thickness. We are allowed to cut the plate, use all or part of it and glue the edges to create a 3D object into which we can pour water (or any liquid).
Cutting the plate into infinitely many tiny pieces to create a bowl may be possible but it is not practical.
The solution that came into my mind is to create a cone by cutting the plate into two pieces of sectors. For a certain angle that can be found with differential calculus, one of the sectors produces a cone with the greatest volume.
Question
Is the cone the only possible solution with the greatest volume? How to prove this problem? I have no idea. Calculus of variation may help? But what function do I have to find to be maximized?
Edit
- In the first step I have two cones. The smaller one can be merged as follows by creating a hole in the bigger one.
A piece of plate obtained by creating the hole above can also be used to create other cones.
Etc...