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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 02 May 2017, 20:02
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) 3√3
(E) 2√2

My 2 cents.
Plug in.

So the area of shaded region needs to be 3 times the area of smaller circular region.
Let's say smaller circular region has r=2, so the area would be 4 pi.
As the area of shaded region needs to be 3 times, then it would be 12 pi.
As the area of shaded region can be found by larger - smaller circles, then we have x - 12pi = 4pi.
Then the x would be 16 pi, which means that its radius is 4.

Therefore, the circumference of big circle is 8 pi and that of smaller circle is 4 pi.
C is the correct answer.
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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 06 Jul 2017, 13:45
Using larger radius as x and smaller one as y, we get : \(πx^{2} - πy^{2} = 3πy^{2}\) OR \(πx^{2} = 4πy^{2}\)............(1)

We need to find the ratio of the circumference, \(\frac{2πx}{2πy}= ?\) OR \(\frac{x}{y}= ?\)

From (1); \(\frac{x}{y} =\frac{2}{1}\)

Hence answer is 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 18 Mar 2018, 23:14
shaded region is 3 three times the area of inner circle.

so, total circle area is 4 times the area of inner circle.

big circle area/small circle area = 4/1

(if two triangles are similiar, if the length of the sides are in ratio, k : 1, the area must be k^2 : 1)
we can extend this concept here, since areas are in ratio 4:1, lengths(radius, diameter, circumference) must be in ratio 2:1

(C)
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In the figure shown, if the area of the shaded region is 3 times  [#permalink]

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New post 27 Mar 2018, 11:00
Assume smaller circle has the radius of 4. The shaded region has an area 3 times the smaller circle.
Smaller = 16pi
Shaded = 48pi
Total circle area = smaller + shadeded = 64pi

64pi = pi*r^2(area of circle)
r= 8 (radius of total circle

circumference - 2*pi*r

Circumference or bigger circle = 2*pi*8 = 16pi
Circuference of smaller circle = 2*pi*4= 8pi

Larger circle circumference is double smaller circle or 2 times the smaller circle
answer 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 29 Apr 2018, 03:59
reto wrote:
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


Plug in some values for the the whole circle. I took r = 4. Which means, the whole circle has area of 16*pi which equals 4x of the area. So the shaded region is 3x, hence 12pi and the small region is 4pi. Therefore the small circle has radius 2 and diameter 4pi.

The large circle has diameter 8pi because i picked initially 4 as the radius (2*r = 8).

Hence the large circumference is double the small one. Answer C.


Hi pushpitkc,

i am trying to understand the above solution

if radius = 4 and the whole circle area is 16*pi how can it be equal to 4x ??

also how can shaded region area be 3x ? somehow cant wrap my mind around this this :)

is it " hence 12pi and the small region is 4pi" 12 pi is it area :? and what about 4pi

morever, when i read question stem " 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?"

i undestand it like this
pi*r^2 = area of circle
2pi *r = circimference of circle

let area of larger non shaded region is 6

then if if the area of the shaded region is 3 times the area of the smaller circular region, then area of smaller is 2, because 2 goes into 6 three times... this is just a first idea solution that came to my mind :)

have a great weekend :)
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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 29 Apr 2018, 12:15
1
dave13 wrote:
reto wrote:
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


Plug in some values for the the whole circle. I took r = 4. Which means, the whole circle has area of 16*pi which equals 4x of the area. So the shaded region is 3x, hence 12pi and the small region is 4pi. Therefore the small circle has radius 2 and diameter 4pi.

The large circle has diameter 8pi because i picked initially 4 as the radius (2*r = 8).

Hence the large circumference is double the small one. Answer C.


Hi pushpitkc,

i am trying to understand the above solution

if radius = 4 and the whole circle area is 16*pi how can it be equal to 4x ??

also how can shaded region area be 3x ? somehow cant wrap my mind around this this :)

is it " hence 12pi and the small region is 4pi" 12 pi is it area :? and what about 4pi

morever, when i read question stem " 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?"

i undestand it like this
pi*r^2 = area of circle
2pi *r = circimference of circle

let area of larger non shaded region is 6

then if if the area of the shaded region is 3 times the area of the smaller circular region, then area of smaller is 2, because 2 goes into 6 three times... this is just a first idea solution that came to my mind :)

have a great weekend :)


Hey dave13

To understand the part about the circle having an area 4x, you will have to
take a look at the question which states that "area of the shaded region is 3
times the area of the smaller circular region"

We are assuming the smaller circle to have area x. Since the shaded region
has 3 times the area, making the area of the shaded region 3x. Now, the area
of the circle which contains the small circle and the shaded region will have an
area of x+3x = 4x

Once that part is clear, hope you understood the flaw in your reasoning!
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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
Re: In the figure shown, if the area of the shaded region is 3 times &nbs [#permalink] 20 May 2018, 03:04

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