Question

a rectangular cardboard measuring 34 in long and 20 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 answer.

Explanation:

Step1: Calculate area of rectangle

The formula for the area of a rectangle is \( A = l \times w \), where \( l = 34 \) in and \( w = 20 \) in.
\( A_{rectangle} = 34 \times 20 = 680 \) square inches.

Step2: Calculate area of semicircle

The diameter of the semicircle is equal to the width of the rectangle, so \( d = 20 \) in, which means the radius \( r = \frac{d}{2} = \frac{20}{2} = 10 \) in.
The formula for the area of a full circle is \( A = \pi r^2 \), so the area of a semicircle is \( \frac{1}{2} \pi r^2 \). Using \( \pi = 3.14 \):
\( A_{semicircle} = \frac{1}{2} \times 3.14 \times 10^2 = \frac{1}{2} \times 3.14 \times 100 = 157 \) square inches.

Step3: Subtract semicircle area from rectangle area

To find the remaining area, subtract the area of the semicircle from the area of the rectangle:
\( A_{remaining} = A_{rectangle} - A_{semicircle} = 680 - 157 = 523 \) square inches.

Answer:

\( 523 \) square inches (or \( 523\ \text{in}^2 \))