SOLUTION: A cube fits exactly inside a sphere and a smaller sphere fits exactly inside the cube. Find the ratio of the volume of the smaller sphere to the volume of the larger sphere.

Question 1178353: A cube fits exactly inside a sphere and a smaller sphere fits exactly inside the cube. Find the ratio of the volume of the smaller sphere to the volume of the larger sphere.
Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

Let r be the radius of the smaller sphere.

Then the edge of the cube is 2r.

The space diagonal of the cube is sqrt(3) times the edge, or 2r*sqrt(3); and it is the diameter of the larger sphere.

So the radius of the larger sphere is r*sqrt(3).

The ratio of the radii of the two spheres is r:r*sqrt(3), or 1:sqrt(3).

The ratio of the volumes is the cube of that ratio.

ANSWER: The ratio of the volume of the smaller sphere to the volume of the larger sphere is 1:3*sqrt(3).

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