I have enough pure silver to coat [math] square meters of surface area. I plan to coat a sphere and a cube, and to use all of the silver in doing so. Allowing for the possibility of all of the silver going onto one of the solids, what dimensions should they be if the total volume of the silvered solids is to be a maximum?

The radius of the sphere is meters,

and the length of the sides of the cube is meters.


Again, allowing for the possibility of all of the silver going onto one of the solids, what dimensions should they be if the total volume of the silvered solids is to be a minimum?

The radius of the sphere is meters,

and the length of the sides of the cube is meters.

Solution:

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