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.

Algebra ->  Volume -> 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.      Log On


   

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) About Me  (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).