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How can you slice a vertically oriented 3D cone to get a 2D parabola?
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19. Conic Sections
Introduction to Conic Sections
Multiple Choice
How can you slice a vertically oriented 3D cone with a 2D plane to get a circle?
A
Slice the cone with a horizontal plane.
B
Slice the cone with a slightly tilted plane.
C
Slice the cone with a heavily tilted plane.
D
Slice the cone with a vertical plane.
검증된 단계별 안내1
Understand the geometry of a cone: A cone is a three-dimensional shape with a circular base and a single vertex. When slicing a cone, the orientation of the plane relative to the cone determines the shape of the cross-section.
Identify the goal: To obtain a circle as the cross-section, the plane must intersect the cone in such a way that the intersection is a perfect circle.
Consider the orientation of the plane: A horizontal plane, which is parallel to the base of the cone, will intersect the cone in a circular cross-section. This is because the plane cuts through the cone at a constant distance from the vertex, maintaining the circular shape.
Explore other orientations: A slightly tilted plane can also produce an ellipse, which is a stretched circle. However, for a perfect circle, the plane must be horizontal.
Visualize the intersection: Imagine slicing through the cone horizontally. The plane will cut through the cone parallel to its base, resulting in a circular cross-section. This is the only orientation that guarantees a perfect circle.
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