To find out what kind of 3-D shape would be created by rotating the given figure around the [tex]\(x\)[/tex]-axis, we first examine the coordinates of the vertices of the figure. The vertices are:
- [tex]\((2, 0)\)[/tex]
- [tex]\((2, -2)\)[/tex]
- [tex]\((6, -2)\)[/tex]
- [tex]\((6, 0)\)[/tex]
These coordinates form a rectangle on the coordinate plane. Let's analyze this step by step.
### Step-by-Step Solution:
1. Understanding Rotation Around the [tex]\(x\)[/tex]-Axis:
When a planar figure is rotated around the [tex]\(x\)[/tex]-axis, every point on the figure traces a circular path in 3-dimensional space. Specifically, the [tex]\(y\)[/tex]-coordinates of the figure determine the radius of these circular paths.
2. Dimensional Analysis:
- Height (along the [tex]\(x\)[/tex]-axis):
The [tex]\(x\)[/tex]-coordinates of the rectangle are [tex]\(2\)[/tex] and [tex]\(6\)[/tex]. Therefore, the length of the rectangle along the [tex]\(x\)[/tex]-axis is the difference between these coordinates:
[tex]\[
\text{Height} = 6 - 2 = 4 \text{ units}
\][/tex]
- Radius (distance from the [tex]\(x\)[/tex]-axis):
The [tex]\(y\)[/tex]-coordinates of the rectangle are [tex]\(0\)[/tex] and [tex]\(-2\)[/tex]. The furthest distance from the [tex]\(x\)[/tex]-axis is the absolute value of the [tex]\(y\)[/tex]-coordinate that is furthest from the axis, which is:
[tex]\[
\text{Radius} = | -2 | = 2 \text{ units}
\][/tex]
3. Shape Determination:
When the rectangle is rotated around the [tex]\(x\)[/tex]-axis, it creates a 3-D shape where each point on the side of the rectangle closest to the axis and furthest from it sweeps out a circular path. The height of the resulting solid is the original length along the [tex]\(x\)[/tex]-axis, and the radius is determined by the maximum absolute [tex]\(y\)[/tex]-coordinate.
In this case, rotating a rectangle along the [tex]\(x\)[/tex]-axis gives rise to a cylindrical shape with a consistent radius.
### Conclusion:
The 3-D shape formed by rotating the rectangle with vertices at [tex]\((2, 0)\)[/tex], [tex]\((2, -2)\)[/tex], [tex]\((6, -2)\)[/tex], and [tex]\((6, 0)\)[/tex] around the [tex]\(x\)[/tex]-axis is a cylinder. The dimensions of this cylinder are:
- Height: 4 units
- Radius: 2 units
Hence, the 3-D shape created is a cylinder with a height of [tex]\(4\)[/tex] units and a radius of [tex]\(2\)[/tex] units.
