Question

a rectangular piece of paper with length 29 cm and width 14 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 \( A_{rectangle} = length \times width \). Given length \( = 29 \, \text{cm} \) and width \( = 14 \, \text{cm} \), so \( A_{rectangle} = 29 \times 14 = 406 \, \text{cm}^2 \).

Step2: Analyze the semicircles

The two semicircles (one at the top, one at the bottom) together form a full circle. The diameter of each semicircle is equal to the width of the rectangle, which is \( 14 \, \text{cm} \). So the radius \( r = \frac{14}{2} = 7 \, \text{cm} \).

Step3: Calculate area of the circle

The formula for the area of a circle is \( A_{circle} = \pi r^2 \). Using \( \pi = 3.14 \) and \( r = 7 \, \text{cm} \), we get \( A_{circle} = 3.14 \times 7^2 = 3.14 \times 49 = 153.86 \, \text{cm}^2 \).

Step4: Calculate remaining area

The remaining area is the area of the rectangle minus the area of the circle (since the two semicircles make a full circle). So \( A_{remaining} = A_{rectangle} - A_{circle} = 406 - 153.86 = 252.14 \, \text{cm}^2 \).

Answer:

\( 252.14 \, \text{cm}^2 \)