Question

The shaded prism below is created from the rectangular box as shown. Points A, B, and C are midpoints of their respective edges. Explain, using Cavalieri's Principle, why the volume of the shaded prism must be one-fourth of the volume of the original rectangular box.
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Answer 1

Answer:

The volume of the prism is 1/4 of the volume of the rectangular box.

Step-by-step explanation:

The figure alluding to the exercise is required, according to the description I will attach the one that must be to be able to solve the exercise.

The first thing is that the cross sections of the prism are triangles and in addition those triangles are congruent to each other with areas equal to the area of ​​the base triangle.

By congruence we can say that the triangle has 1/4 of the area of ​​the base rectangle. We can affirm that the height of the prism is equal to the height of the rectangular box.

Now, the Cavalieri principle states that if two solids have the same height and their cross-sectional areas taken parallel and at equal distances from their bases are always equal, then they have the same volume.

now in this case the cross-sectional areas (parallel to the base) of the prism and the cross-sectional areas (parallel to the base) of the cuboid with a height equal to that of the rectangular box and the length, width of half of the sizes of the rectangular box are always the same.

Which means that the volume of the parallelepiped is 1/4 the volume of the rectangular box and thanks to this we can say that the volume of the prism is 1/4 of the volume of the rectangular box.