SOLUTION: the cube of largest volume is cut out from a sphere of radius 4 root 3. find the volume of the cube

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Question 935703: the cube of largest volume is cut out from a sphere of radius 4 root 3. find the volume of the cube
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Intuition tells me this has a very expected symmetry to use. The distance between any two opposite corners of the cube must be a diameter of the sphere.

That distance is found with the Pythagorean Theorem in THREE DIMENSIONS. If assign s = length of edge of the cube,
s%5E2%2Bs%5E2%2Bs%5E2=%282%284sqrt%283%29%29%29%5E2.

3s%5E2=4%2A16%2A3
s%5E2=4%2A16
s=2%2A4
highlight%28s=8%29, the edge length of the cube.

The volume of this cube is highlight%28s%5E3=8%5E3=highlight%28512%29%29.