Question

if a rectangular cross - section with coordinates at (1,1),(1,4),(3,4) and (3,1) is rotated about the line y = 1, what will be the resulting three - dimensional object?
cone
cylinder
sphere
pyramid

Explanation:

Step1: Analyze the shape of the cross - section

The cross - section with coordinates \((1,1)\), \((1,4)\), \((3,4)\) and \((3,1)\) is a rectangle.

Step2: Consider the rotation about the line \(y = 1\)

When a rectangle is rotated about a line parallel to one of its sides (in this case, the line \(y=1\) is parallel to the side of the rectangle with endpoints \((1,1)\) and \((3,1)\)), we can use the method of generating a 3D - object from a 2D - shape.
The height of the rectangle (distance between \(y = 1\) and \(y = 4\)) is \(h=4 - 1=3\), and the length of the side perpendicular to the axis of rotation (distance between \(x = 1\) and \(x = 3\)) is \(r = 3 - 1=2\).
When a rectangle is rotated about a line parallel to one of its sides, the resulting 3 - dimensional object is a cylinder. The formula for the volume of a cylinder is \(V=\pi r^{2}h\), where \(r\) is the radius (distance from the axis of rotation to the side of the rectangle perpendicular to the axis) and \(h\) is the height (length of the side parallel to the axis of rotation). Here \(r = 2\) and \(h=3\).

Answer:

Cylinder