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Question 4 (Essay Worth 10 points)

A figure is located at [tex]\((0,0)\)[/tex], [tex]\((-3,-4)\)[/tex], and [tex]\((-3,0)\)[/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 :

GinnyAnswer
To determine the 3-D shape formed by rotating a 2-D figure around the x-axis, we start by examining the given coordinates of the points on the figure: [tex]\((0,0)\)[/tex], [tex]\((-3,-4)\)[/tex], and [tex]\((-3,0)\)[/tex].

### Step-by-Step Solution:

1. Plotting the Points:
- The point [tex]\((0,0)\)[/tex] is the origin.
- The point [tex]\((-3,-4)\)[/tex] is located 3 units to the left along the x-axis and 4 units down along the y-axis.
- The point [tex]\((-3,0)\)[/tex] is located 3 units to the left along the x-axis.

2. Connecting the Points:
- Connect [tex]\((0,0)\)[/tex] to [tex]\((-3,0)\)[/tex] forming a horizontal line along the x-axis.
- Connect [tex]\((-3,0)\)[/tex] to [tex]\((-3,-4)\)[/tex] forming a vertical line along the y-axis.

This creates a right triangle with [tex]\((0,0)\)[/tex], [tex]\((-3,0)\)[/tex], and [tex]\((-3,-4)\)[/tex].

3. Rotation Around the x-axis:
- When the triangle is rotated around the x-axis, each point on the figure will trace a circular path perpendicular to the x-axis.
- The line segment from [tex]\((0,0)\)[/tex] to [tex]\((-3,0)\)[/tex] will create a cylindrical surface.
- The vertical line from [tex]\((-3,0)\)[/tex] to [tex]\((-3,-4)\)[/tex] will sweep out a conical shape with its base on the y-plane and its height extending down the z-axis.

4. Resulting 3-D Shape Analysis:
- Cylinder: This cylinder has a height extending from [tex]\(x=0\)[/tex] to [tex]\(x=-3\)[/tex]. Hence, the height of the cylinder is 3 units.
- Cone (Frustum): The rotation of the vertical segment creates a conical shape. Since the vertical line segment's length (distance from y=0 to y=-4) is 4 units, the radius of this revolving shape is 4 units.

### Conclusion:
The 3-D shape resulting from this rotation is a combination of a cylinder and a conical frustum. Specifically, the:
- Height of the cylinder is 3 units (from [tex]\(x=0\)[/tex] to [tex]\(x=-3\)[/tex]).
- Radius of the cone (and thus the conical surface) is 4 units (from [tex]\(y=0\)[/tex] to [tex]\(y=-4\)[/tex]).

Therefore, the detailed dimensions of this combined 3-D shape are a height of 3 units for the cylinder and a radius of 4 units for the conic section.
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