Answered

Experience the convenience of getting your questions answered at IDNLearn.com. Receive prompt and accurate responses to your questions from our community of knowledgeable professionals ready to assist you at any time.

Question 4 (Essay, Worth 10 Points)

A figure is located at [tex]\((2,0)\)[/tex], [tex]\((2,-2)\)[/tex], [tex]\((6,-2)\)[/tex], and [tex]\((6,0)\)[/tex] on a coordinate plane. What kind of 3-D shape would be created if the figure was rotated around the [tex]\(x\)[/tex]-axis? Provide an explanation and proof of your answer to receive full credit. Include the dimensions of the 3-D shape in your explanation.

Sagot :

GinnyAnsweridn
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
We are delighted to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Thank you for choosing IDNLearn.com for your queries. We’re here to provide accurate answers, so visit us again soon.