Question

a figure is located at (2, 0), (2, -2), and (6, 0) on a coordinate plane. what kind of 3 - d shape would be created if the figure was rotated around the x - axis? provide an explanation and proof of your answer to receive full credit. include the dimensions of the 3 - d shape in your explanation.

Explanation:

Step1: Analyze the 2 - D shape

The points (2,0), (2, - 2), and (6,0) form a right - triangle in the xy - plane. The base of the triangle lies on the x - axis from x = 2 to x = 6 (length of base $b=4$) and the height of the triangle is from y = 0 to y=-2 (height $h = 2$).

Step2: Recall the solid of revolution concept

When a 2 - D shape is rotated about the x - axis, we can use the disk/washer method in calculus (or geometric reasoning for simple shapes). For a right - triangle rotated about the x - axis, the resulting 3 - D shape is a cone.

Step3: Justify the cone formation

As we rotate the right - triangle around the x - axis, each cross - section perpendicular to the x - axis is a circle. The radius of these circles varies linearly from 0 (at the vertex of the triangle on the x - axis) to 2 (at the non - x - axis vertex of the triangle). This is characteristic of a cone, where the radius of circular cross - sections changes linearly from the apex to the base.

Answer:

A cone would be created. The right - triangle formed by the points (2,0), (2, - 2), and (6,0) when rotated about the x - axis generates a cone because the cross - sections perpendicular to the axis of rotation (x - axis) are circles with radii that vary linearly, which is a defining feature of a cone.