SOLUTION: A wooden pyramid of altitude h is to be sawed into three parts of equal weight. How far from the vertex must the cuts (parallel to the base) be made?

Question 1193436: A wooden pyramid of altitude h is to be sawed into three parts of equal weight. How far from the vertex must the cuts (parallel to the base) be made?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The weight is proportional to the volume; the distance from the vertex is a linear measurement. So, given similar figures, the ratio of heights is the cube root of the ratio of volumes.

To get a pyramid with 1/3 the volume of the whole pyramid, the height of the pyramid needs to be (cube root of 1/3) times the volume of the whole pyramid:

= 0.69336 to 5 decimal places.

To get a pyramid with 2/3 the volume of the whole pyramid, the height of the pyramid needs to be (cube root of 2/3) times the volume of the whole pyramid:

= 0.87538 to 5 decimal places.

ANSWER: To get three pieces with the same weight, the two planes cutting the pyramid of height h parallel to the base should be at distances 0.69336h and 0.87538h from the vertex.

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