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Select all solids whose cross sections are dilations of some two- dimensional shape using a point directly above the shape as a center and scale factors ranging from 0 to 1. Cylinder Cube Triangular Prism Cone Triangular Pyramid

User Wellington Souza
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1 Answer

24 votes
24 votes

Ok, so:

Let's look at the cone:

The cone has a relation with the equation of dilating the cylinder by 1/3, as this:

V cone = (1/3) * pi (r)^2 * h

V cylinder = pi * (r)^2 * h

Then, the cross-section of the cylinder being dilated by 1/3 (which is between 0 and 1) gives the cone.

So, cone is a solid whose cross sections are dilations of some two- dimensional shape.

Ok, then, the volume of a triangular prism is given by:

V = 0.5 * b * a * h, where h is ths height, b and a are its sides.

The volume of a triangular pyramid is given by:

V = 1/3 ( Base area * height).

If we do not know the Base area, we can write it like b*a.

So, these two equations has a relation, as this:

(0.5 b*a*h) / (1/3 b*a*h) = 1.5

This means that if we multiply the volume of the triangular pyramid by 1/1.5, the result is 0.5

Then, the cross-section of the triangular pyramid being dilated by 1/1.5 (which is between 0 and 1) gives the triangular prism.

So, the answer would be Cone and triangular prism.

User Giwyni
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