Question

A rectangular paperboard measuring 29 in long and 22 in wide has a semicircle cut out of it, as shown below.
Find the area of the paperboard that remains. Use the value 3.14 for pi, and do not round your answer. Be sure to include the correct unit in your answer.

Answer 1

To find the area of the remaining paperboard after cutting out a semicircle, we first need to determine the area of the rectangular paperboard and the area of the semicircle.

1. **Area of the Rectangular Paperboard**:
The area \( A_{rectangle} \) of a rectangle is given by the formula:
\[
A_{rectangle} = \text{length} \times \text{width}
\]
For the paperboard:
\[
A_{rectangle} = 29 \, \text{in} \times 22 \, \text{in} = 638 \, \text{in}^2
\]

2. **Area of the Semicircle**:
The area \( A_{semicircle} \) of a semicircle can be calculated using the formula:
\[
A_{semicircle} = \frac{1}{2} \pi r^2
\]
To find the radius \( r \), we assume the semicircle is cut from the width of the rectangular paperboard (22 inches). Therefore, the radius \( r \) is half the width:
\[
r = \frac{22 \, \text{in}}{2} = 11 \, \text{in}
\]

Now, we can calculate the area of the semicircle:
\[
A_{semicircle} = \frac{1}{2} \times 3.14 \times (11 \, \text{in})^2
\]
\[
A_{semicircle} = \frac{1}{2} \times 3.14 \times 121 \, \text{in}^2
\]
\[
A_{semicircle} = 1.57 \times 121 \, \text{in}^2 = 189.97 \, \text{in}^2
\]

3. **Remaining Area**:
To find the area of the paperboard that remains after the semicircle is cut out, we subtract the area of the semicircle from the area of the rectangular paperboard:
\[
A_{remaining} = A_{rectangle} - A_{semicircle}
\]
\[
A_{remaining} = 638 \, \text{in}^2 - 189.97 \, \text{in}^2
\]
\[
A_{remaining} = 448.03 \, \text{in}^2
\]

Thus, the area of the paperboard that remains is **448.03 in²**.