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In the figure shown, if the area of the shaded region is 3 times
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mimajit
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Re: In the figure shown, if the area of the shaded region is 3 times [#permalink] New post  20 May 2018, 02:04
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.

Show: ::
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] New post  15 Sep 2019, 09:39
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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.

Show: ::
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] New post  17 Sep 2019, 00:00
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] New post  02 May 2020, 09:42
[Area of Bigger circle ][/Area of smaller circle]=SQUARE of [Radius of bigger circle][/Radius of smaller circle].

Area of smaller circle=1 & Radius of smaller circle=1.
Difference between Area of bigger circle & Area of smaller circle=3, So, Area of bigger circle=4. So, Radius of bigger circle= 2.


[Circumference of Bigger circle ][/Circumference of smaller circle]= [Radius of bigger circle][/Radius of smaller circle].

Radius of smaller circle=1 & Radius of bigger circle= 2.


TAKEAWAY
AREA RATIO=SQUARE of RADIUS RATIO.
CIRCUMFERENCE RATIO= RADIUS RATIO.
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Re: In the figure shown, if the area of the shaded region is 3 times [#permalink] New post  19 Jul 2020, 22:55
Let,

small circle area be 1 PI
Big circle be 3 PI as three times

Hence total circle area is 4 PI

Circumference of Big circle.

4 PI = R^2 * pi
therefore R =2 and circumference is 4PI

Circumference of small circle.

1PI = PI*R^2

R = 1, hence circumference is 2PI

hence 2 is answer
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Re: In the figure shown, if the area of the shaded region is 3 times [#permalink] New post  18 Aug 2020, 19:34
Rule for Similar Triangles applies for Circles as well, because Every Circle is Similar to Every Other Circle

Ratio of ----- Area of Inner Smaller Circle : Area of ENTIRE Circle = 1 : 4

then the Ratio of ----- Radius of Inner Smaller Circle : Radius of ENTIRE Circle = Square Root of Each Part of Ratio = 1 : 2

Using r = 1 for Inner Small Circle and R = 2 for Entire Outer Circle

Circumference of Inner Circle = 2*pi
Circumference of Entire Circle = 4*pi

Answer C - the Circumference of Outer Circle is 2 TIMES the Circumference of Inner Smaller Circle
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In the figure shown, if the area of the shaded region is 3 times [#permalink] New post  06 Jan 2021, 13:13
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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) \(\sqrt{2}\)

Show: ::
Attachment:
2015-10-22_0821.png


Let, The radius of the larger circle \(= R\), and the radius of the smaller circle \(= r\)

So, the Area of the larger circle \(= πR^2\), and the are of the smaller circle \(= πr^2\)

Given that,
\(πR^2-πr^2=3πr^2\)

\(⇒πR^2=4πr^2\)

\(⇒R^2=4r^2\)

\(⇒R=\sqrt{4r^2}\)

\(⇒R=2r\)

The circumference of the larger circle \(= 2πR\), and the circumference of the smaller circle \(=2πr\)

The comparison: \(\frac{2πR}{2πr}=\frac{2π*2r}{2πr}=2\)

The answer is \(C\)
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Re: In the figure shown, if the area of the shaded region is 3 times [#permalink] New post  09 Jan 2021, 08:59
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Very simple way to solve this problem

Assume : Smaller circle area is 9
Then : Shaded region area is 3 times the area of smaller region therefore -> 27
Therefore : Whole circle area is 36 ( Adding the smaller and the shaded region)

Now radius of whole circle is 6 (Since area is 36)
and radius of smaller circle is 3 (Since area is 9)

Therefore the circumference of the bigger circle is 2* the circumference of the smaller circle.
2*6*pi -> bigger circle & 2*3*pi smaller circle
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In the figure shown, if the area of the shaded region is 3 times [#permalink] New post  09 Jan 2021, 11:51
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Luciano wrote:
Very simple way to solve this problem

Assume : Smaller circle area is 9
Then : Shaded region area is 3 times the area of smaller region therefore -> 27
Therefore : Whole circle area is 36 ( Adding the smaller and the shaded region)

Now radius of whole circle is 6 (Since area is 36)
and radius of smaller circle is 3 (Since area is 9)

Therefore the circumference of the bigger circle is 2* the circumference of the smaller circle.
2*6*pi -> bigger circle & 2*3*pi smaller circle


I like your way of solving, and shaped it.

Let, Smaller circle area is 9,

Then the Shaded region area is \(= 9*3=27\) [Three times the area of the smaller region]

Area of the larger circle (whole circle) \(= 9+27=36\)

∴ The are of the larger circle \(πR^2=36, \ or \ πr=6\), or The perimeter \(2πR=12\)

The are of the smaller circle \(πr^2=9, \ or \ πr=3\), or The perimeter \(2πr=6\)

\( \frac{The \ perimeter \ of \ the \ larger \ circle \ 2πR =12}{ The \ perimeter \ of \ the \ smaller \ circle \ 2πr =6}=2\)

The answer is \(C\)
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