Bunuel wrote:

If the area of the shaded region in the figure above is 24π, what is the radius r of the smaller circle?

(A) 2

(B) 4

(C) 5

(D) 6

(E) 10

Attachment:

2017-11-29_0805_001.png

Shaded region's area =

\(24\pi\)Small circle's area =

\(\pi r^2\)Large circle's area =

\(\pi (r + 2)^2\)Area of the shaded region =

(Large circle's area) - (Small circle's area)

\(\pi (r + 2)^2 - (\pi r^2) = 24\pi\)\(\pi (r^2 + 4r + 4) - (\pi r^2) = 24\pi\)\(\pi r^2 + 4\pi r + 4\pi - \pi r^2 = 24\pi\)\(4\pi r + 4\pi = 24\pi\)\(4\pi r = 20\pi\)\(r = 5\)Answer C

Check:

Small circle's area, with r = 5, = \(25 \pi\)

Large circle's area, with r = 7, = \(49\pi\)

Large - Small = shaded region

\(49\pi - 25\pi = 24\pi\). That works.
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