Question

creating three - dimensional solids
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:

Part 1: Three - dimensional solid created

When a rectangle is rotated about one of its sides (line \(k\) here), the three - dimensional solid formed is a cylinder. This is based on the concept of solid of revolution, where a rectangle rotated about an axis (the side parallel to one of its sides) forms a cylinder.

Step 1: Recall the formula for the perimeter of a rectangle

The perimeter of a rectangle \(P = 2(l + w)\), where \(l\) is the length and \(w\) is the width. We know that the length of the rectangle (the side parallel to line \(k\)) \(l=10\) units and \(P = 32\) units.

Step 2: Solve for the width of the rectangle

Substitute the known values into the perimeter formula:
\[32=2(10 + w)\]
First, divide both sides by 2: \(\frac{32}{2}=10 + w\), so \(16=10 + w\).
Then, subtract 10 from both sides: \(w=16 - 10=6\) units.
When the rectangle is rotated about line \(k\) (the side of length 10), the width of the rectangle becomes the radius of the base of the cylinder.

Answer:

Cylinder

Part 2: Radius of the base