The right circular cone above is sliced horizontally forming two pieces, each of which has the same height. What is the ratio of the volume of the smaller piece to the volume of the larger piece?
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It is helpful to label the figure.
The top piece is a cone whose height is one-half the height of the original cone . Using the properties of similar right triangles, you should realize the radii of these two cones must be in the same ratio. So if the top cone has radius , the original cone has radius .
The volume of the top piece is equal to . The volume of the bottom piece is equal to the volume of the original cone minus the volume of the top piece.
The ratio of the volume of the smaller piece to the volume of the larger piece is .