Question

a rectangular paperboard measuring 33 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 π, and do not round your answer. be sure to include the correct unit in your answer.

Explanation:

Step1: Calculate the area of the rectangle

The area formula for a rectangle is \(A = l\times w\). Here, \(l = 33\) in and \(w = 22\) in.
\(A_{rectangle}=33\times22 = 726\) \(in^{2}\)

Step2: Calculate the area of the semicircle

The diameter of the semicircle \(d = 22\) in, so the radius \(r=\frac{d}{2}=\frac{22}{2}=11\) in. The area formula for a semicircle is \(A=\frac{1}{2}\pi r^{2}\). Using \(\pi = 3.14\), we have \(A_{semicircle}=\frac{1}{2}\times3.14\times11^{2}=\frac{1}{2}\times3.14\times121 = 189.97\) \(in^{2}\)

Step3: Calculate the remaining area

Subtract the area of the semicircle from the area of the rectangle. \(A = A_{rectangle}-A_{semicircle}\)
\(A=726 - 189.97=536.03\) \(in^{2}\)

Answer:

\(536.03\) \(in^{2}\)