Let's carefully analyze the process of rotating a square about a line and determine the properties of the resulting solid.
Consider a square with side length [tex]\( s \)[/tex]. When this square is rotated about one of its sides, we get a solid of revolution.
To identify the resulting solid, imagine rotating the square about one of its sides:
1. Rotating a square about one of its sides:
When the square rotates about one of its sides, it sweeps out a three-dimensional space.
This movement creates a cylinder. The height of this cylinder equals the side length of the square, and the radius of the cylinder also equals the side length of the square. The reason it's a cylinder and not a cube is due to the shape traced, which involves circular motion ending up forming cylindrical bases.
So far, we have established:
- The solid formed by rotating the square is a cylinder.
Now let's analyze each statement to verify its truth:
A) The solid is a cube.
- False. Rotating a square about its side does not produce a cube. A cube would have equal dimensions in all three directions, but this isn't the case here.
B) The solid is a rectangular prism.
- False. A rectangular prism has faces that are rectangles, not circles. This solid is not a rectangular prism.
C) The solid is a cylinder.
- True. As established, the rotation of a square about one of its sides results in a cylinder.
D) The height of the solid is twice the width of the solid.
- False. The height of the cylinder is equal to the side of the square, and the diameter is also twice the side length of the square. However, considering "width" as a dimension in a different direction might not apply accurately here.
E) The diameter of the solid is twice the height of the solid.
- True. The height of the solid (cylinder) is [tex]\( s \)[/tex], and the diameter (twice the radius) is [tex]\( 2s \)[/tex], which indeed is twice the height.
F) The diameter of the solid is equal to the height of the solid.
- False. The diameter of the cylinder is twice its height.
Based on the analysis:
The true statements are:
- C) The solid is a cylinder.
- E) The diameter of the solid is twice the height of the solid.
