In the diagram above are two concentric circles of radii r and R respectively with…

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

Correct Answer: Option 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