SOLUTION: Show that the rectangular solid of maximum volume that can be inscribed in a sphere is a cube?

Question 389078: Show that the rectangular solid of maximum volume that can be inscribed in a sphere is a cube?
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Without loss of generality, let the diameter of the sphere be (you'll see why I picked such a strange number), and that the dimensions of the rectangular solid are x, y, z. Given that:

we want to show that for all positive x, y, z that

(where 1 is the equality case and the volume of an inscribed cube).

Using the AM-GM inequality,

Since , this becomes

Cubing both sides,

, as desired. Note that the equality case is where x = y = z = 1, which is also the equality case of AM-GM.

There is probably an optimization solution involving derivatives, however, it would involve several variables and would be fairly tedious. I'm sure many other solutions exist. This solution is the first one that came to my mind.
RELATED QUESTIONS
Show that the rectangular solid of maximum surface area inscribed in a sphere is a... (answered by richard1234)

Find the volume of the largest sphere that can be inscribed in a cube of edge 12... (answered by stanbon)

Determine the dimensions of the cylinder of maximum volume that can be inscribed in a... (answered by solver91311)

In a cube, 10" on a side, a sphere of maximum radius is inscribed. What is the size... (answered by RAY100)

a rectangular solid of maximum volume is to be cut from a solid sphere of radius... (answered by ccs2011)

If a rectangular solid (4x5x7) is inscribed in a sphere, what is the radius of the... (answered by ankor@dixie-net.com)

A cube with volume 8 cubic centimeters is inscribed in a sphere so that each vertex of... (answered by scott8148)

A sphere of radius r is inscribed in a cube. What is the ratio of the volume of the... (answered by Alan3354)

a sphere of radius r is inscribed in a cube. what is the ratio of the volume of the... (answered by Alan3354)