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Sagot :
In this problem, we have a rectangle located on a coordinate plane with vertices at the points [tex]\((2,0)\)[/tex], [tex]\((2,-2)\)[/tex], [tex]\((6,-2)\)[/tex], and [tex]\((6,0)\)[/tex]. We are asked to determine the 3-dimensional shape created when this rectangle is rotated around the [tex]\(x\)[/tex]-axis, and to provide the dimensions of that shape.
### Step-by-Step Solution:
1. Understand the Geometry of the Given Coordinates:
- The vertices [tex]\((2,0)\)[/tex], [tex]\((2,-2)\)[/tex], [tex]\((6,-2)\)[/tex], and [tex]\((6,0)\)[/tex] form a rectangle on the x-y plane.
- The distance between [tex]\((2,0)\)[/tex] and [tex]\((2,-2)\)[/tex] (or between [tex]\((6,0)\)[/tex] and [tex]\((6,-2)\)[/tex]) is 2 units (height of the rectangle along the y-axis).
- The distance between [tex]\((2,-2)\)[/tex] and [tex]\((6,-2)\)[/tex] (or between [tex]\((2,0)\)[/tex] and [tex]\((6,0)\)[/tex]) is 4 units (width of the rectangle along the x-axis).
2. Analyze the Rotation Around the [tex]\(x\)[/tex]-axis:
- When we rotate a shape around the [tex]\(x\)[/tex]-axis, points with the same x-coordinate trace out a circle around the axis.
- In this case, the vertical sides of the rectangle that span from [tex]\(y = 0\)[/tex] to [tex]\(y = -2\)[/tex] will form circles with a radius equal to the y-distance, which is 2 units.
- The horizontal sides (spanning [tex]\(x = 2\)[/tex] to [tex]\(x = 6\)[/tex]) will translate into the height of the cylindrical shape.
3. Determine the Resulting 3-Dimensional Shape:
- The rectangle rotation forms a cylinder.
- The height of the cylinder is the same as the width of the rectangle, which is 4 units.
- The radius of the cylinder is the height of the rectangle, which is 2 units.
### Dimensions of the Cylinder:
- Radius: Radius is the distance from the x-axis to either [tex]\(y = -2\)[/tex] or [tex]\(y = 0\)[/tex]. This distance is 2 units.
- Height: The height of the cylinder is equal to the width of the rectangle, which is 4 units.
### Conclusion:
When the rectangle with vertices at [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 forms a cylinder. The dimensions of this cylinder are:
- Height: 4 units
- Radius: 2 units
### Step-by-Step Solution:
1. Understand the Geometry of the Given Coordinates:
- The vertices [tex]\((2,0)\)[/tex], [tex]\((2,-2)\)[/tex], [tex]\((6,-2)\)[/tex], and [tex]\((6,0)\)[/tex] form a rectangle on the x-y plane.
- The distance between [tex]\((2,0)\)[/tex] and [tex]\((2,-2)\)[/tex] (or between [tex]\((6,0)\)[/tex] and [tex]\((6,-2)\)[/tex]) is 2 units (height of the rectangle along the y-axis).
- The distance between [tex]\((2,-2)\)[/tex] and [tex]\((6,-2)\)[/tex] (or between [tex]\((2,0)\)[/tex] and [tex]\((6,0)\)[/tex]) is 4 units (width of the rectangle along the x-axis).
2. Analyze the Rotation Around the [tex]\(x\)[/tex]-axis:
- When we rotate a shape around the [tex]\(x\)[/tex]-axis, points with the same x-coordinate trace out a circle around the axis.
- In this case, the vertical sides of the rectangle that span from [tex]\(y = 0\)[/tex] to [tex]\(y = -2\)[/tex] will form circles with a radius equal to the y-distance, which is 2 units.
- The horizontal sides (spanning [tex]\(x = 2\)[/tex] to [tex]\(x = 6\)[/tex]) will translate into the height of the cylindrical shape.
3. Determine the Resulting 3-Dimensional Shape:
- The rectangle rotation forms a cylinder.
- The height of the cylinder is the same as the width of the rectangle, which is 4 units.
- The radius of the cylinder is the height of the rectangle, which is 2 units.
### Dimensions of the Cylinder:
- Radius: Radius is the distance from the x-axis to either [tex]\(y = -2\)[/tex] or [tex]\(y = 0\)[/tex]. This distance is 2 units.
- Height: The height of the cylinder is equal to the width of the rectangle, which is 4 units.
### Conclusion:
When the rectangle with vertices at [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 forms a cylinder. The dimensions of this cylinder are:
- Height: 4 units
- Radius: 2 units
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