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 21/25πR^2
  • B 9/25πR^2
  • C 21/23πR^2
  • D 5/9πR^2
jamb 2002mathematics

The correct answer is D. 5/9πR^2

r = \(\frac{2}{3}\)R

∴R = \(\frac{3}{3}\)R

Area of small circle = πr

= π(\(\frac{2R}{3}\))

Area of the big circle πr

= π\(\frac{(3R)^2}{3}\)

Area of shaded portion = π(\(\frac{3R}{3}\))

- π(\(\frac{2R}{3}\))

= π[(\(\frac{3R}{3}\))

- (\(\frac{2R}{3}\))

]

= 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 \(\frac{R}{3}\)

=5/9πR^3

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