In the diagram, the perimeter of the semicircular region is 20 . (The perimeter includes both the semicircular arc and the diameter.) The area of the region is closest to
(A) 36.6
(B) 23.8
(C) 49.3
(D) 51.6
(E) 26.7

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-088.jpg?height=201&width=355&top_left_y=702&top_left_x=1292)
Suppose that the radius of the region is $r$.

The length of the semi-circle is half of the circumference, or $\frac{1}{2}(2 \pi r)=\pi r$.

Thus, the perimeter of the shaded region is $\pi r+2 r$.

Since the perimeter is 20 , then $\pi r+2 r=20$ or $r(\pi+2)=20$ or $r=\frac{20}{\pi+2}$.

Thus, the area of the semi-circle is half of the area of a circle with this radius, or $\frac{1}{2} \pi\left(\frac{20}{\pi+2}\right)^{2}$ which is approximately 23.768, which of the given choices is closest to 23.8 .

ANswer: (B)