Question

a rectangular cardboard measuring 20 in long and 12 in wide has a semicircle cut out of it, as shown below. what is the perimeter of t cardboard that remains after the semicircle is removed? (use the value 3.14 for π, and do not round your answer. be sure to include the unit in your answer.)

Explanation:

Step1: Analyze the perimeter components

The remaining perimeter consists of two lengths of the rectangle, one width, and the length of the semicircular arc. The diameter of the semicircle is equal to the width of the rectangle, 12 in, so the radius \( r = \frac{12}{2}=6 \) in. The formula for the circumference of a full circle is \( C = 2\pi r \), so the length of the semicircular arc is \( \frac{1}{2} \times 2\pi r=\pi r \).

Step2: Calculate each component

• Lengths of the rectangle: There are two lengths, each 20 in, so total length contribution is \( 2\times20 = 40 \) in.
• Width of the rectangle: One width, 12 in.
• Semicircular arc length: Using \( \pi = 3.14 \) and \( r = 6 \) in, the arc length is \( 3.14\times6 = 18.84 \) in.

Step3: Sum the components

Add the three parts together: \( 40 + 12 + 18.84 \).
First, \( 40+12 = 52 \), then \( 52 + 18.84 = 70.84 \) in.

Answer:

70.84 inches