Question

a rectangular piece of paper with length 34 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 = 34 cm and width = 14 cm, so \( A_{rectangle} = 34 \times 14 \).
\( 34 \times 14 = 476 \) \( cm^2 \)

Step2: Calculate area of the two semicircles (which form a circle)

The two semicircles have the same diameter as the width of the rectangle, so diameter \( d = 14 \) cm, radius \( r = \frac{d}{2} = \frac{14}{2} = 7 \) cm. The area of a circle is \( A_{circle} = \pi r^2 \), with \( \pi = 3.14 \). So \( A_{circle} = 3.14 \times 7^2 \).
First, calculate \( 7^2 = 49 \), then \( 3.14 \times 49 = 153.86 \) \( cm^2 \)

Step3: Subtract the area of the circle from the area of the rectangle

The remaining area \( A = A_{rectangle} - A_{circle} \). So \( A = 476 - 153.86 \).
\( 476 - 153.86 = 322.14 \)

Answer:

\( 322.14 \) square centimeters