To determine the 3-D shape created by rotating the given figure around the [tex]$x$[/tex]-axis, we first need to understand the figure on the coordinate plane.
### Step-by-Step Analysis:
1. Identify the Coordinates and Shape:
The vertices provided are:
- [tex]\( (2,0) \)[/tex]
- [tex]\( (2,-2) \)[/tex]
- [tex]\( (6,-2) \)[/tex]
- [tex]\( (6,0) \)[/tex]
Plotting these points on the coordinate plane, we can see that they form a rectangle.
2. Visualizing the Rotation:
When we rotate a 2-dimensional figure around an axis, it generates a 3-dimensional shape. Here, we are rotating around the [tex]$x$[/tex]-axis.
3. Dimensions of the Rectangle:
- The height of the rectangle is determined by the difference in the [tex]$y$[/tex]-coordinates: [tex]\( 0 - (-2) = 2 \)[/tex]
- The width of the rectangle is determined by the difference in the [tex]$x$[/tex]-coordinates: [tex]\( 6 - 2 = 4 \)[/tex]
4. Resultant 3-D Shape:
By rotating the rectangle around the [tex]$x$[/tex]-axis, we generate a 3-dimensional cylindrical shape because each vertical line gets swept into a circle.
- The length of the cylinder is the width of the rectangle, which is [tex]\( 4 \)[/tex] units (the distance between [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex]).
- The radius of the cylinder is the height of the rectangle, which is [tex]\( 2 \)[/tex] units (the distance from [tex]\( y = 0 \)[/tex] to [tex]\( y = -2 \)[/tex]).
5. Calculating the Volume of the Cylinder:
To confirm the shape and further describe its properties, let's determine its volume.
- Radius ([tex]\( r \)[/tex]) = 2 units
- Height ([tex]\( h \)[/tex]) = 4 units
The formula for the volume of a cylinder is given by:
[tex]\[ V = \pi r^2 h \][/tex]
Plugging in the values:
[tex]\[ V = \pi (2)^2 (4) \][/tex]
[tex]\[ V = \pi (4) (4) \][/tex]
[tex]\[ V = 16\pi \][/tex]
So, the volume approximates to:
[tex]\[ V \approx 50.27 \text{ cubic units} \][/tex]
### Conclusion:
When the rectangle with vertices [tex]\( (2,0) \)[/tex], [tex]\( (2,-2) \)[/tex], [tex]\( (6,-2) \)[/tex], and [tex]\( (6,0) \)[/tex] is rotated around the [tex]$x$[/tex]-axis, it creates a cylindrical shape. The cylinder has a radius of [tex]\( 2 \)[/tex] units and a height of [tex]\( 4 \)[/tex] units. The volume of the cylinder is approximately [tex]\( 50.27 \)[/tex] cubic units.
