Question

a rectangular paperboard measuring 32 in long and 26 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 rectangle area

The area formula for a rectangle is $A_{rect}=l\times w$, where $l = 32$ in and $w = 26$ in. So $A_{rect}=32\times26=832$ in².

Step2: Calculate semicircle area

The diameter of the semicircle is equal to the width of the rectangle, so $d = 26$ in, and the radius $r=\frac{d}{2}=\frac{26}{2}=13$ in. The area formula for a full - circle is $A_{circle}=\pi r^{2}$, and for a semi - circle $A_{semicircle}=\frac{1}{2}\pi r^{2}$. Substituting $r = 13$ in and $\pi=3.14$, we get $A_{semicircle}=\frac{1}{2}\times3.14\times13^{2}=\frac{1}{2}\times3.14\times169 = 265.33$ in².

Step3: Calculate remaining area

The remaining area $A = A_{rect}-A_{semicircle}$. So $A=832 - 265.33=566.67$ in².

Answer:

$566.67$ in²