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# In the figure shown, if the area of the shaded region is 3

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Director
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In the figure shown, if the area of the shaded region is 3 [#permalink]

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04 Feb 2006, 16:52
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In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?

A. $$4$$
B. $$3$$
C. $$2$$
D. $$\sqrt{3}$$
E. $$\sqrt{2}$$

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-the-figure-shown-if-the-area-of-the-shaded-region-is-3-t-104668.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 31 Oct 2013, 03:55, edited 4 times in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.
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04 Feb 2006, 16:56
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Two times?

Smaller circle = pi*r^2
Larger circle = pi*R^2

Given ,

pi*R^2 - pi*r^2 = 3* pi*r^2

R^2 = 4*pi*r^2

R = 2*r

Ratio of circimference = 2*pi*R/2*pi*r = 2
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04 Feb 2006, 17:32
Agree with Giddi's working
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Director
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04 Feb 2006, 17:43
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I think it should be 1+ sqrt3 or 1 - sqrt3

Area of the shaded region is : Pi(R-r)^2 = 3 Pi r^2
(R-r)^2 = 3r^2
R-r =rsqrt3
or
R-r = -rsqrt3

If i See the options available for the PS, I might be able to answer it..

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Re: In the figure shown, if the area of the shaded region is 3 [#permalink]

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31 Oct 2013, 03:09
Let small circle radius = a
Area of shaded region = PIE (a+b)^2 - PIE a^2 = PIE b (2a+b).............. (i)
Area of shaded region is thrice that of Area of smaller circle
So, 3 PIE a^2 = PIE b (2a+b)

3a^2 - 2ab - b^2 = 0
Solving above quadratic equation gives a = b
It means radius of bigger circle is twice than radius of smaller circle

Circumference = 2 PIE (radius) means
Circumference of bigger circle is twice than circumference of smaller circle
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Re: In the figure shown, if the area of the shaded region is 3 [#permalink]

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31 Oct 2013, 03:55
Expert's post

In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?

A. $$4$$
B. $$3$$
C. $$2$$
D. $$\sqrt{3}$$
E. $$\sqrt{2}$$

The area of the shaded region is $$area_{shaded}=\pi{R^2}-\pi{r^2}$$ and the area of the smaller circle is $$area_{small}=\pi{r^2}$$. Given: $$\pi{R^2}-\pi{r^2}=3\pi{r^2}$$ --> $$R^2=4r^2$$ --> $$R=2r$$;

Now, the ratio of the circumference of the larger circle to the that of the smaller circle is $$\frac{C}{c}=\frac{2\pi{R}}{2\pi{r}}=\frac{{2r}}{{r}}=2$$.

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-the-figure-shown-if-the-area-of-the-shaded-region-is-3-t-104668.html
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Re: In the figure shown, if the area of the shaded region is 3   [#permalink] 31 Oct 2013, 03:55
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