Question

practice 6 (from unit 5, lesson 3)
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.
a cylinder
b cube
c triangular prism
d cone
e triangular pyramid

Explanation:

To solve this, we analyze each solid's cross - sections:

• Cylinder (A): Cross - sections parallel to the base are circles of the same size (scale factor = 1), not dilations with scale factors from 0 to 1 (excluding 1 in the range we need for non - constant dilation).
• Cube (B): Cross - sections parallel to a face are squares of the same size (scale factor = 1), so not what we want.
• Triangular Prism (C): Cross - sections parallel to the base are triangles of the same size (scale factor = 1), not dilations with scale factors from 0 to 1.
• Cone (D): A cone has a circular base. When we take a cross - section parallel to the base at a height \(h\) from the base (and \(H\) is the height of the cone), the radius \(r\) of the cross - section circle and the radius \(R\) of the base circle are related by \(\frac{r}{R}=\frac{H - h}{H}\), which is a scale factor ranging from 0 (at the apex) to 1 (at the base). So the cross - sections are dilations of the circular base with a center (the apex) directly above the base.
• Triangular Pyramid (E): A triangular pyramid (tetrahedron) has a triangular base. When we take a cross - section parallel to the base, the triangle of the cross - section and the base triangle are similar. The scale factor of the dilation (with the apex as the center) ranges from 0 (at the apex) to 1 (at the base).

Answer:

D. cone, E. triangular pyramid