Carcass wrote:
Each of the three shaded regions above is a semicircle. If AB = 4, CD = 2BC, and BC = 2AB, then the area of the entire shaded figure is
A. 28π
B. 42π
C. 84π
D. 96π
E. 168π
From the given information, we get:
The
diameter of the smallest semicircle is
4, which means its
radius =
2 The
diameter of the middle semicircle is
8, which means its
radius =
4 The
diameter of the largest semicircle is
16, which means its
radius =
8 Area of semicircle \(= \frac{\pi r^2}{2}\)So, the area of the smallest semicircle
\(= \frac{\pi (2^2)}{2} = 2 \pi\)The area of the middle semicircle
\(= \frac{\pi (4^2)}{2} = 8 \pi\)The area of the largest smallest semicircle
\(= \frac{\pi (8^2)}{2} = 32 \pi\)So, the TOTAL shaded area
\(= 2 \pi + 8 \pi + 32 \pi = 42 \pi \)Answer: B