A figure is located at (2, 0), (2, −2), (6, −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.

Answer 1

To understand what 3-D shape is formed by rotating the given figure around the x-axis, let's first analyze the coordinates provided: (2, 0), (2, -2), (6, -2), and (6, 0).

Figure Description

The points (2, 0) and (2, -2) correspond to the left edge of a vertical line segment, while the points (6, 0) and (6, -2) correspond to the right edge of another vertical line segment. Connecting these points, we can see that we have a rectangle that is vertical on the coordinate plane with:

• Left corner at (2, 0)
• Left corner at (2, -2)
• Right corner at (6, 0)
• Right corner at (6, -2)

This rectangle has a width of \(6 - 2 = 4\) units and a height of \(0 - (-2) = 2\) units.

Rotation around the x-axis

When this rectangle is rotated around the x-axis, each point on the vertical sides of the rectangle will trace out a circle. The rotation will create two cylindrical surfaces:

1. From (2, 0) to (2, -2): The vertical line at \(x = 2\) will produce a cylinder of radius 2 units (since y ranges from 0 to -2) when rotated around the x-axis.
2. From (6, 0) to (6, -2): The vertical line at \(x = 6\) will produce another cylinder of radius 2 units.

Dimensions of the 3-D shape

Given the dimensions:

• Height of each cylinder: The height of the rectangle is \(2\) units.
• Radius of each cylinder: The radius is \(2\) units.

Shape Produced

The figure will create a 3-D shape composed of a cylindrical layer with a radius of 2 units extending from x = 2 to x = 6, resulting in a larger cylindrical volume with the following properties:

• Height: The height of the cylinder when viewed along the x-axis will be \(4\) units (the width we derived earlier).
• Overall Radius: Since both vertical edges produce the same height of 2 around the corresponding centers, the overall structure can be seen as a cylindrical volume with outer radius at x = 6 and inner radius at x = 2.

Conclusion

The result of this rotation around the x-axis creates a cylindrical shape with:

• Outer Radius: 2 units (for the section from x = 2).
• Inner Radius: 4 units (for the section from x = 6).
• Length along x-axis: \(6 - 2 = 4\) units.

Thus, the final shape is a hollow cylinder (or hollow tube) extending from \(x = 2\) to \(x = 6\) with a radius of \(2\) units.