Optimization puzzle: Cone containing a unit sphere
I'd like to think I made up this puzzle, but given the simplicity of the puzzle statement, I'm sure it's been conceived many times before.
While this GeoGebra construction may be used to help visualize the premise, the puzzle is intended to be solved without technology. Calculus required unless someone more clever than myself can find non-calculus solutions.
In the construction below, the 2D view shows the base of the cone with radius r (disk) and the unwrapped lateral side of the cone (sector). The 3D view shows the cone and sphere, and a button toggles between the 3D representation and its 2D cross section.
Values of r may either be entered in the input box, or the blue point on the edge of the cone may be dragged in the 2D/3D views.
Puzzle questions are posed in the graphic above. Solutions are not provided here, although you may check your answers for correctness. Your puzzle answers may be entered in the labeled input boxes below. Either exact expressions or decimal approximations may be entered. If an answer is correct to three decimal places, the corresponding gray ✘ will convert to a green ✔.
When is the volume of the cone minimized? The lateral surface area? The total surface area?
Even when exploring different values of r above, it's hard to tell.