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:

Identify the 2-D figure

The given vertices are \(A(2, 0)\), \(B(2, -2)\), and \(C(6, 0)\).
Plotting these points reveals a right triangle.
The right angle is located at vertex \(A(2, 0)\).
The legs are along \(x = 2\) and the x-axis.

Determine the dimensions

The horizontal leg \(AC\) lies on the x-axis.
Its length is \(6 - 2 = 4\) units.
The vertical leg \(AB\) lies on the line \(x = 2\).
Its length is \(|0 - (-2)| = 2\) units.

Rotate around the x-axis

Rotating a right triangle about one of its legs creates a cone.
Since the rotation axis is the x-axis, the leg \(AC\) lies on the axis of rotation.
The leg \(AB\) is perpendicular to the axis of rotation.
This perpendicular leg \(AB\) sweeps out the circular base.

Find the 3-D shape dimensions

The radius \(r\) of the cone's base is the length of leg \(AB\).
Thus, \(r = 2\) units.
The height \(h\) of the cone is the length of leg \(AC\).
Thus, \(h = 4\) units.
The vertex of the cone is at \((6, 0)\).
The center of the circular base is at \((2, 0)\).

Answer:

A cone with a radius of 2 units and a height of 4 units.