To solve this problem, let's visualize what happens when rotating a rectangular cross-section about the line [tex]\(y = 1\)[/tex].
1. Identify the given coordinates: The given rectangular cross-section has vertices at:
- [tex]\((1, 1)\)[/tex]
- [tex]\((1, 4)\)[/tex]
- [tex]\((3, 4)\)[/tex]
- [tex]\((3, 1)\)[/tex]
2. Position observation:
- Note that the line [tex]\(y = 1\)[/tex] lies along the bottom line (y-axis) of the rectangle.
3. Visualize the rotation:
- When the rectangle is rotated around [tex]\(y = 1\)[/tex], it will sweep out a three-dimensional object.
- The sides [tex]\((1, 1)\)[/tex] to [tex]\((1, 4)\)[/tex] and [tex]\((3, 1)\)[/tex] to [tex]\((3, 4)\)[/tex], which are vertical relative to the rotation axis, will generate cylindrical surfaces.
- The top edge at [tex]\((1, 4)\)[/tex] to [tex]\((3, 4)\)[/tex] will form circular end caps.
4. Understand the resulting shape:
- The rotation of the rectangular cross-section around the horizontal line produces a solid object with circular symmetry around the rotation axis.
- In this case, it forms a cylindrical shape with a height determined by the rectangular cross-section’s width and the radius determined by the height of the rectangle from [tex]\(y = 1\)[/tex] to [tex]\(y = 4\)[/tex].
So, the correct answer is:
O Cylinder
