Question

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a rectangular paperboard measuring 30 in long and 20 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 of a rectangle is given by \(A = length\times width\). Here, \(length = 30\) in and \(width=20\) in. So, \(A_{rectangle}=30\times20 = 600\) \(in^{2}\).

Step2: Calculate the area of the semicircle

The diameter of the semicircle \(d = 20\) in, so the radius \(r=\frac{d}{2}=\frac{20}{2}=10\) in. The area of a full - circle is \(A=\pi r^{2}\), and the area of a semicircle is \(A_{semicircle}=\frac{1}{2}\pi r^{2}\). Substituting \(\pi = 3.14\) and \(r = 10\) in, we get \(A_{semicircle}=\frac{1}{2}\times3.14\times10^{2}=\frac{1}{2}\times3.14\times100 = 157\) \(in^{2}\).

Step3: Calculate the remaining area

The remaining area \(A = A_{rectangle}-A_{semicircle}\). Substitute \(A_{rectangle}=600\) \(in^{2}\) and \(A_{semicircle}=157\) \(in^{2}\) into the formula. So, \(A=600 - 157=443\) \(in^{2}\).

Answer:

\(443\) \(in^{2}\)