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How can you slice a vertically oriented 3D cone with a 2D plane to get a circle?
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19. Conic Sections
Introduction to Conic Sections
Multiple Choice
How can you slice a vertically oriented 3D cone to get a 2D parabola?
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.
Consider the orientation of the plane: To obtain a parabola, the plane must be tilted relative to the axis of the cone. A horizontal plane will produce a circle, while a vertical plane will produce a hyperbola.
Visualize the slicing process: Imagine a plane that is tilted at an angle, not parallel to the base and not perpendicular to the axis of the cone. This plane should intersect the cone in such a way that it cuts through both the curved surface and the base.
Determine the angle of tilt: The plane should be heavily tilted, meaning it is angled enough to intersect the cone in a way that the cross-section forms a parabola. This requires the plane to be neither too shallow nor too steep.
Recognize the resulting shape: When the plane is heavily tilted, the intersection with the cone will form a parabola. This is because the curve of the cone and the angle of the plane create a parabolic shape in the cross-section.
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