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.

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the geometry of a cone. A cone is a three-dimensional shape with a circular base that tapers smoothly to a point called the apex. When slicing a cone, the intersection of the plane and the cone determines the resulting 2D shape.
2. Step 2: Recall the conic sections. Conic sections are the curves obtained by intersecting a cone with a plane. These include circles, ellipses, parabolas, and hyperbolas, depending on the angle and orientation of the slicing plane.
3. Step 3: To obtain a parabola, the slicing plane must be tilted at an angle such that it is parallel to the slant height of the cone. This ensures the intersection forms a parabola, which is defined as a curve where every point is equidistant from a fixed point (focus) and a fixed line (directrix).
4. Step 4: A horizontal plane will produce a circle, while a slightly tilted plane will produce an ellipse. A heavily tilted plane will produce a hyperbola. To achieve a parabola, the plane must be tilted at just the right angle relative to the cone's axis.
5. Step 5: Visualize the slicing process. Imagine the cone and the plane intersecting. The parabola is formed when the plane cuts through one nappe (half) of the cone and is parallel to the cone's slant height. This geometric relationship is key to understanding why the resulting shape is a parabola.