Question

a rectangular piece of paper with length 27 cm and width 12 cm has two semicircles cut out of it, as shown below. find the area of the paper 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 \( A = l \times w \), where \( l = 27 \, \text{cm} \) and \( w = 12 \, \text{cm} \).
\( A_{\text{rectangle}} = 27 \times 12 = 324 \, \text{cm}^2 \)

Step2: Analyze the two semicircles

The two semicircles together form a full circle. The diameter of each semicircle is equal to the width of the rectangle, \( d = 12 \, \text{cm} \), so the radius \( r = \frac{d}{2} = \frac{12}{2} = 6 \, \text{cm} \).

Step3: Calculate area of the circle

The formula for the area of a circle is \( A = \pi r^2 \). Using \( \pi = 3.14 \) and \( r = 6 \, \text{cm} \):
\( A_{\text{circle}} = 3.14 \times 6^2 = 3.14 \times 36 = 113.04 \, \text{cm}^2 \)

Step4: Find remaining area

Subtract the area of the circle from the area of the rectangle:
\( A_{\text{remaining}} = A_{\text{rectangle}} - A_{\text{circle}} = 324 - 113.04 = 210.96 \, \text{cm}^2 \)

Answer:

\( 210.96 \, \text{square centimeters} \) (or \( 210.96 \, \text{cm}^2 \))