The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to its base. If its volume is \frac{1}{27} of the volume of the given cone, then what is the height of the frustum of the cone?

Explanation

Let the height of the small cone be h. The ratio of their volumes is the cube of the ratio of their heights: (\frac{h}{30})^3 = \frac{1}{27} \implies \frac{h}{30} = \frac{1}{3} \implies h = 10\text{ cm}. The height of the frustum is 30 - 10 = 20\text{ cm}.

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