Problem 22
<h3>Answers:</h3>
- vertical: rectangle
- horizontal: rectangle
- angled: parallelogram
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Explanation:
The vertical and horizontal cross sections are fairly straight forward. They are simply mirror images of the outward showing faces. The angled cross sections are a bit more complicated and there's a lengthy proof involved, but long story short, the angled cutting plane divides the 3D solid such that we have 2 sets of lines that have the same slope (if we consider a 2D view), which leads to 2 sets of parallel sides.
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Problem 23
<h3>Answers:</h3>
- vertical: either a triangle or quadrilateral
- horizontal: triangle
- angled: either a triangle or quadrilateral
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Explanation:
The horizontal cross section is always a triangle because the bottom base face is a triangle. The other two types of cross sections are either triangles or quadrilaterals depending on where the cutting plane is situated. For vertical cross sections that go through the apex point, we get a triangle. For vertical cross sections that do not go through the apex, then we get a quadrilateral. Sometimes a trapezoid is possible here, but not always. It's better to just consider it a quadrilateral to be the most general. A similar situation happens with the angled cuts as well.
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Problem 24
<h3>Answers:</h3>
- vertical: triangle, but only if plane is crossing through apex
- horizontal: circle
- angled: ellipse or parabola
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Explanation:
Imagine you shined a flashlight onto the cone such that the flashlight is perfectly level and flat. It would cast a shadow that is a triangle. This is one way to think of a cross section. If you vertically slice. The horizontal cross sections are always circles due to the circular base of the cone. The angled cross sections are either ellipses or parabolas. For more information, look in your math textbook about conic sections (just ignore the second cone however).
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Problem 25
<h3>Answers:</h3>
- vertical: rectangle
- horizontal: circle
- angled: ellipse
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Explanation:
The horizontal cross sections are circles for similar reasoning as the cone horizontal cross section. However, this time the vertical cross sections are rectangles. The widest possible rectangle is the result of the vertical cutting plane passing through the center of the circular base. Angled cross sections are ellipses. Though some portions of the ellipse may be cut off depending on what the actual angle is.
