Multiple choice

A right pyramid with a square base has side of base 12 cm and height 40 cm. It is kept on its base. It is cut into four parts of equal height by three cuts parallel to its base. What is the ratio of the volumes of the four parts?

  1. 1 : 8 : 27 : 70

  2. 1 : 7 : 19 : 47

  3. 1 : 7 : 19 : 37

  4. 1 : 8 : 27 : 64
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C Correct answer
Explanation

The pyramid is cut into 4 parts of equal height. The volumes of the pyramids formed by the top 1, 2, 3, and 4 sections follow the ratio of the cubes of their heights: 1^3 : 2^3 : 3^3 : 4^3 = 1 : 8 : 27 : 64. The volumes of the individual frustum parts are 1, (8-1)=7, (27-8)=19, and (64-27)=37.

AI explanation

When a pyramid is cut by planes parallel to the base, the volumes of the similar pyramids from the apex to each cut are proportional to the cubes of their height ratios. The heights are in the ratio 1 to 2 to 3 to 4, so the volumes of the top sections are proportional to 1 cubed, 2 cubed, 3 cubed, and 4 cubed, which are 1, 8, 27, and 64. The volumes of the four individual parts are found by subtracting the volume of the section above it from each total, yielding 1, 8 minus 1, 27 minus 8, and 64 minus 27. This results in the ratio 1 to 7 to 19 to 37.

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