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This problem trains for: SAT-I, AMC-8, GMAT, SAT-II.

A small sphere is inside a larger sphere. If the volume of the little sphere varies quadratically with the radius of the large sphere, by what factor does the volume of the large sphere increase if the radius of the small sphere increases 2-fold?

The volume of a sphere varies cubically with the radius of the sphere:

equation

Denote the radius of the large sphere with RL.

Therefore, if the radius of the small sphere increases 2-fold, then its volume increases 8-fold.

The volume of the small sphere is:

equation

equation

and becomes:

equation

equation

The new volume of the large sphere:

equation

equation

equation

equation