Question

rectangle abcd has a perimeter of 32 units and will be rotated about line k what is the three - dimensional solid created? what is the radius of a base?

Explanation:

Step1: Determine the three - dimensional solid

When a rectangle is rotated about a line, the three - dimensional solid created is a cylinder.

Step2: Find the width of the rectangle

The perimeter formula of a rectangle is \(P = 2(l + w)\), where \(P = 32\) and \(l=10\).
Substitute into the formula: \(32=2(10 + w)\)
Divide both sides by 2: \(16 = 10+w\)
Subtract 10 from both sides: \(w=6\)

Step3: Determine the radius of the base of the cylinder

When the rectangle is rotated about line \(k\), the side of the rectangle adjacent to line \(k\) (the length of the rectangle) forms the radius of the base of the cylinder. So the radius \(r = 10\)

Answer:

1. Cylinder
2. \(10\)