Cone- smaller cone, volume of 270cm^3, height of small cone is two thirds of the original.

A cone has a volume of 270cm^3. A smaller cone is cut off the top. The height of the new cone is two-thirds of the original cone. What volume of the material is left after the smaller cone is removed?
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The thing about this question is we don't need to know how to find the volume of a cone. The reason is that the smaller cone and larger cone are similar. It also means that the heights alone will determine the ratios of their volumes. Height is a single dimension and volume is three-dimensional, so the ratio of the heights has to be cubed when considering the ratios of the volumes. Let's call the height of the smaller cone h and that of the larger one H. We do the same with the volumes v and V. Since h/H=2/3, v/V= (2/3)^3=8/27 and we know V=270cc so v=8/27*270=80cc. The volume of remaining material is 270-80=190cc.

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