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O and O' are the centres of the circles of radius $r$ as shown in figures (i) and (ii) respectively.
Find the ratio of area of shaded region in figure (i) to that of area of shaded region in figure (ii).
Find the ratio of area of shaded region in figure (i) to that of area of shaded region in figure (ii).
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$217.14$ cm (1/2 Mark)
OR
(B) Area of unshaded segments in figure (i) $= 2 (\frac{60}{360} \pi r^2 - \frac{\sqrt{3}}{4} r^2)$ (1 Mark)
Getting length of rectangle $= \sqrt{3}r$ and width $= r$ (1 Mark)
Area of rectangle $= \sqrt{3}r^2$ (1 Mark)
Area of shaded region in figure (i) $= \pi r^2 - 2 (\frac{60}{360} \pi r^2 - \frac{\sqrt{3}}{4} r^2)$ (1 Mark)
$= \frac{2}{3} \pi r^2 - \frac{\sqrt{3}}{2} r^2$ (1/2 Mark)
Area of shaded region in figure (ii) $= 2 (\frac{60}{360} \pi r^2 - \frac{\sqrt{3}}{4} r^2) = \frac{1}{3} \pi r^2 - \frac{\sqrt{3}}{2} r^2$ (1 Mark)
Required ratio $= \frac{\frac{2}{3} \pi r^2 - \frac{\sqrt{3}}{2} r^2}{\pi r^2 - (\frac{1}{3} \pi r^2 - \frac{\sqrt{3}}{2} r^2)} = \frac{4\pi-3\sqrt{3}}{2\pi-3\sqrt{3}}$ (1/2 Mark)
or $(4\pi - 3\sqrt{3}): (2\pi - 3\sqrt{3})$
or $(88 - 21\sqrt{3}): (44 - 21\sqrt{3})$
OR
(B) Area of unshaded segments in figure (i) $= 2 (\frac{60}{360} \pi r^2 - \frac{\sqrt{3}}{4} r^2)$ (1 Mark)
Getting length of rectangle $= \sqrt{3}r$ and width $= r$ (1 Mark)
Area of rectangle $= \sqrt{3}r^2$ (1 Mark)
Area of shaded region in figure (i) $= \pi r^2 - 2 (\frac{60}{360} \pi r^2 - \frac{\sqrt{3}}{4} r^2)$ (1 Mark)
$= \frac{2}{3} \pi r^2 - \frac{\sqrt{3}}{2} r^2$ (1/2 Mark)
Area of shaded region in figure (ii) $= 2 (\frac{60}{360} \pi r^2 - \frac{\sqrt{3}}{4} r^2) = \frac{1}{3} \pi r^2 - \frac{\sqrt{3}}{2} r^2$ (1 Mark)
Required ratio $= \frac{\frac{2}{3} \pi r^2 - \frac{\sqrt{3}}{2} r^2}{\pi r^2 - (\frac{1}{3} \pi r^2 - \frac{\sqrt{3}}{2} r^2)} = \frac{4\pi-3\sqrt{3}}{2\pi-3\sqrt{3}}$ (1/2 Mark)
or $(4\pi - 3\sqrt{3}): (2\pi - 3\sqrt{3})$
or $(88 - 21\sqrt{3}): (44 - 21\sqrt{3})$