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

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

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New post 20 May 2018, 03:04
Bunuel wrote:
Image
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) \(\sqrt{2}\)

Kudos for a correct solution.

Attachment:
2015-10-22_0821.png




This is how I solved. Assume the radius of smaller circle r to be 2. then the area = PI (r)^2= Pi (2)^2 = 4 PI
Its also told that area of shaded region is 3 times the smaller circle = 12 PI
Now smaller circle + Shaded area = area of larger circle = 4Pi + 12Pi = 16 Pi
from this we can get the radius of Larger circle = Pi (R)^2 = 16 pi
cancelling pi from both sides = (R)^2 = 16 therefore R=4
Now that we know R we can find out Circumference of Larger circle = 2 PI R = 8 PI

Similarly we know the radius of smaller circle = 2 then Circumference of smaller circle = 4Pi
So to answer the main question that the circumference of the larger circle is how many times the circumference of the smaller circle?
= 8Pi / 4pi = 2 times = C
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Re: In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 15 Sep 2019, 10:39
Bunuel wrote:
Image
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) \(\sqrt{2}\)

Kudos for a correct solution.

Attachment:
2015-10-22_0821.png


Given: In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region.

Asked: The circumference of the larger circle is how many times the circumference of the smaller circle?

Let the radius of larger circle be R and radius of smaller circle be r

Area of shaded region = \(\pi (R^2 - r^2)\)
Area of smaller circle = \(\pi r^2\)

\(\pi (R^2 - r^2) = 3\pi r^2\)
R^2 = 4 r^2
R = 2r

2\pi R = 2 * 2\pi r
The circumference of the larger circle is 2 times the circumference of the smaller circle.

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

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New post 17 Sep 2019, 01:00
A quick confirmation - for circumference there is no subtraction of inner from outer circle required unlike that in area calculation? Thanks
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Re: In the figure shown, if the area of the shaded region is 3 times   [#permalink] 17 Sep 2019, 01:00

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