Question

a rectangular cardboard measuring 20 in long and 12 in wide has a semicircle cut out of it, as shown below. find the area of the cardboard that remains. use the value 3.14 for π, and do not round your answer. be sure to include the correct unit in your answer

Explanation:

Step1: Calculate area of rectangle

The formula for the area of a rectangle is \( A_{\text{rectangle}} = \text{length} \times \text{width} \). Given length \( = 20 \) in and width \( = 12 \) in, so \( A_{\text{rectangle}} = 20 \times 12 = 240 \) \( \text{in}^2 \).

Step2: Calculate radius of semicircle

The diameter of the semicircle is equal to the width of the rectangle, so diameter \( d = 12 \) in. The radius \( r = \frac{d}{2} = \frac{12}{2} = 6 \) in.

Step3: Calculate area of semicircle

The formula for the area of a full circle is \( A_{\text{circle}} = \pi r^2 \), so the area of a semicircle is \( A_{\text{semicircle}} = \frac{1}{2} \pi r^2 \). Substituting \( \pi = 3.14 \) and \( r = 6 \) in, we get \( A_{\text{semicircle}} = \frac{1}{2} \times 3.14 \times 6^2 = \frac{1}{2} \times 3.14 \times 36 = 56.52 \) \( \text{in}^2 \).

Step4: Calculate remaining area

Subtract the area of the semicircle from the area of the rectangle: \( A_{\text{remaining}} = A_{\text{rectangle}} - A_{\text{semicircle}} = 240 - 56.52 = 183.48 \) \( \text{in}^2 \).

Answer:

\( 183.48 \) square inches