To find the area of the shaded region, let's analyze the given diagram.
As we can see, it can be calculated as the difference of the area of the rectangle and the area of the semicircle. Let's calculate each area one at a time.
Area of the Rectangle
The can be determined using the following formula.
\begin{aligned}
A_\text{rectangle}=w\ell
\end{aligned}
Here, w is the width and l is the length of the rectangle. From the diagram we know both these values. Let's substitute w with 4 and l with 6 into the formula.
A_\text{rectangle}=w\ell
A_\text{rectangle}=({\color{#0000FF}{4}})({\color{#009600}{6}})
A_\text{rectangle}=24
Area of the Semicircle
A is one half of a circle, so its area is equal to one-half of the area of the circle with the same . Hence, we can use the following formula.
\begin{aligned}
A_\text{semicircle}=\dfrac{1}{2}\pi r^2
\end{aligned}
We know that the diameter of the semicircle is 4 inches. By dividing 4 by 2, we get that the equals 2 inches. Let's substitute r with 2 and calculate A_\text{semicircle}.
A_\text{semicircle}=\dfrac{1}{2}\pi r^2
A_\text{semicircle}=\dfrac{1}{2}\pi ({\color{#0000FF}{2}})^2
A_\text{semicircle}=\dfrac{1}{2}\pi (4)
A_\text{semicircle}=2\pi
A_\text{semicircle}=6.283185\ldots
A_\text{semicircle}\approx 6.3
Area of the Shaded Region
Now that we know the areas of the rectangle and the semicircle, we can find their difference. This way we can calculate the area of the shaded region.
\begin{aligned}
A_\text{shaded}&=A_\text{rectangle}-A_\text{semicircle} \\
&\Downarrow \\
A_\text{shaded}&=24-6.3=17.7 \text{ in}^2
\end{aligned}
Therefore, the area of the shaded area is about 17.7 square inches.
