A right circular metallic cone of height $20$ cm and radius of base $5$ cm is melted and recast into a sphere. Find the radius of the sphere.

Because the volume remains unchanged when the cone is recast, we equate the volume of the cone, which is given by the formula one third times pi times radius squared times height, to the volume of the sphere, which is four thirds times pi times the new radius cubed. Substituting the given values gives one third times pi times five squared times twenty equals four thirds times pi times the new radius cubed. Canceling the common terms and solving yields twenty-five times twenty equals four times the new radius cubed, so five hundred equals four times the new radius cubed. This gives the new radius cubed as one hundred twenty-five, making the radius of the sphere 5 cm.

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