Write an expression in factored form for the shaded portion in the diagram.
a.
The area of the rectangle is equal to $A_{\text{rectangle}} = LW$, where $L = 4r$ and $W = 2r$. So,
$A_{\text{rectangle}} = 4r(2r) = 8r^2$
And the area of the two circles is
$
\begin{equation}
\begin{aligned}
A_{\text{circle}} =& 2 (\pi r^2)
\\
=& 2 \pi r^2
\end{aligned}
\end{equation}
$
Then, by subtracting the area of the rectangle to the area of circle, we get the area of the shaded portion as
$
\begin{equation}
\begin{aligned}
A_{\text{rectangle}} - A_{\text{circle}} =& 8r^2 - 2 \pi r^2
\\
=& 2r^2 (4 - \pi)
\end{aligned}
\end{equation}
$
b.
Based from the figure, the area of the square is $A_{\text{rectangle}} = (2r)^2 = 4r^2$ and the area of the circle is $A_{\text{circle}} = \pi r^2$
Then, by subtracting the area of the square to the area of the circle, we obtain the area of the shaded portion as
$
\begin{equation}
\begin{aligned}
A_{\text{rectangle}} - A_{\text{circle}} =& 4r^2 - \pi r^2
\\
=& r^2(4 - \pi)
\end{aligned}
\end{equation}
$
Friday, March 22, 2019
Beginning Algebra With Applications, Chapter 8, 8.1, Section 8.1, Problem 130
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