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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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New post 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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New post 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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New post 04 Feb 2006, 17:32
Agree with Giddi's working :wink:
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New post 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..

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

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New post 31 Oct 2013, 03:09
Let small circle radius = a
Big circle radius = (a+b)
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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New post 31 Oct 2013, 03:55
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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. \(\sqrt{2}\)

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 \(\frac{C}{c}=\frac{2\pi{R}}{2\pi{r}}=\frac{{2r}}{{r}}=2\).

Answer: C.

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

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