In the diagram above are two concentric circles of radii r and R respectively with center O. If r = 2/3R, express the area of the shaded portion in terms of π and R
A.21/25πR2 B.9/25πR2 C.21/23πR2 D.5/9πR2
Correct Answer: D
Explanation
r = \(\frac{2}{3}\)R −R = \(\frac{3}{3}\)R Area of small circle = πr2 = π(\(\frac{2R}{3}\))2 Area of the big circle πr2 = π\(\frac{(3R)^2}{3}\) Area of shaded portion = π(\(\frac{3R}{3}\))2 - π(\(\frac{2R}{3}\))2 = π[(\(\frac{3R}{3}\))2 - (\(\frac{2R}{3}\))2] = π[(\(\frac{3R}{3}) + (\frac{2R}{3}) - (\frac{3R}{3}\)) - (\(\frac{2R}{3}\))] = π[(\(\frac{5R}{3}\)) (\(\frac{R}{3}\))] = π x \(\frac{5R}{3}\) x \(\frac{R}{3}\) = 5/9πR2
