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

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

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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. \(\sqrt{2}\)

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Originally posted by monirjewel on 11 Nov 2010, 08:09.
Last edited by Bunuel on 06 Dec 2017, 11:24, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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Re: In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 11 Nov 2010, 08:27
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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.
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Re: In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 11 Nov 2010, 08:27
1
circular shaded region = area of large circle - area of small circle = 3 * area of small circle
so area of large circle = 4 * area of small circle

that means radius oflarge circle = 2 * radius of small circle
so the cicumference of large circle is 2 times that of the small circle ---Option C
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Re: In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 06 Mar 2014, 01:27
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Re: In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 09 Mar 2014, 03:10
2
Let the area of small circle = x

Area of shaded region = 3x

Total Area = x+3x = 4x

This means that Area of larger circle is 4 times Area of Small Circle;

which also means radius of larger circle is 2 times radius of small circle & so the circumference

So Answer = C = 2
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Re: In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 13 Apr 2014, 08:24
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. \(\sqrt{2}\)

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.



HI Bunnel,

Have one doubt on this as you have mentioned here R = 2r. Now we put this in the pi*rsquare then for both circles the are we got is

4pir square = pi r square.

Now area is four times instead of 3 times. So please clarify this
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Re: In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 13 Apr 2014, 08:37
pawankumargadiya 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. \(\sqrt{2}\)

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.



HI Bunnel,

Have one doubt on this as you have mentioned here R = 2r. Now we put this in the pi*rsquare then for both circles the are we got is

4pir square = pi r square.

Now area is four times instead of 3 times. So please clarify this


The stem says that the area of the shaded region is 3 times the area of the smaller circular region, not that the area of the larger circular region is 3 times the area of the smaller circular region.

Does this make sense?
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Re: In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 06 Dec 2017, 11:09
1
monirjewel wrote:
Attachment:
Untitled2.png
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. \(\sqrt{2}\)
Attachment:
1111220830.jpg

Algebraically
Let a = Small circle's area
Let s = Shaded region's area
Let A = Large circle's area

\(s = 3a\)
\(A = s + a\)
\(A = 3a + a\)
\(A = 4a\)

\(a = \pi r^2\)
\(A = \pi R^2\)


From above:
\(A = 4a\)
\(\pi R^2 = 4\pi r^2\)
\(R^2 = 4r^2\)
\(\sqrt{R^2}=\sqrt{4r^2}\)
\(R = 2r\)
*

Circumference, Large and Small
C = \(2\pi R\), and c = \(2\pi r\)
\(R = 2r\), so
\(\frac{C}{c}=\frac{2\pi (2r)}{2\pi r}= \frac{2r}{r}=2\)

The large circle's circumference is two times the small circle's circumference.

Answer C

Numbers and algebra

Large circle's area: Small circle's area?

Small circle's area = x
Shaded region's area = 3x
Large circle's area = 3x + x = 4x

Large circle circumference/ small circle's circumference?

Let Small circle's radius \(r = 1\)
Area of Small: \(\pi r^2 = \pi\)
Area of Large = (4x) = \((4*\pi) = 4\pi\)

Radius, R, of Large circle:
\(4\pi =\pi R^2\)
\(R = 2\)


Circumference, Small: \(2 \pi r = 2 \pi\)
Circumference, Large: \(2 \pi R = 4 \pi\)

\(\frac{Large}{Small}=\frac{4\pi}{2\pi} = 2\)

The large circle's circumference is two times the small circle's circumference.

Answer C

**OR
\(A = 3a + a\)
\(A - 3a = a\) AND \(A = \pi R^2\)

\(\pi R^2 - 3\pi r^2=\pi r^2\)
\(\pi (R^2 - 3r^2)=\pi r^2\)
\(R^2 - 3r^2 = r^2\)
\(R^2 = 4r^2\) Take square roots:
\(R = 2r\)
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In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 25 Dec 2017, 02:19
Bunuel,

The area of the shaded region is areashaded=πR2−πr2areashaded=πR2−πr2 and the area of the smaller circle is areasmall=πr2areasmall=πr2.

I made a mistake here.
I write this like: πR^2=3*πr^2 (as is say 3 times)

What mistake have I made?
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Re: In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 25 Dec 2017, 02:24
amitpandey25 wrote:
Bunuel,

The area of the shaded region is areashaded=πR2−πr2areashaded=πR2−πr2 and the area of the smaller circle is areasmall=πr2areasmall=πr2.

I made a mistake here.
I write this like: πR^2=3*πr^2 (as is say 3 times)

What mistake have I made?


I think it's addressed above.

The stem says that the area of the shaded region is 3 times the area of the smaller circular region, not that the area of the larger circular region is 3 times the area of the smaller circular region.
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Re: In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 01 Feb 2018, 12:19
Hi All,

This question can be solved by TESTing VALUES.

We're told that the area of the shaded region is 3 TIMES the area of the central circle...

Area of center = 1
Area of shaded region = 3(1) = 3
Area of FULL CIRCLE = 1+3 = 4

With those values....
Radius of center = 1
Radius of FULL CIRCLE = 2

The question asks how many times the circumference of the full circle is to the smaller circle...

Circumference of small circle = 2pi
Circumference of full circle = 4pi

Final Answer:

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

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New post 27 Feb 2018, 23:03
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. \(\sqrt{2}\)

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.




Hi Bunuel,

Can you share problems that are similar to this?

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

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New post 27 Feb 2018, 23:15
1
shivamtibrewala 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. \(\sqrt{2}\)

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.




Hi Bunuel,

Can you share problems that are similar to this?

Thanks.


Shaded Region Problems from our Special Questions Directory.
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Re: In the figure shown, if the area of the shaded region is 3 t  [#permalink]

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New post 01 Mar 2018, 17:35
monirjewel 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. \(\sqrt{2}\)

Attachment:
1111220830.jpg
Attachment:
Untitled2.png


If we let A = the radius of the larger circle and B = the radius of the smaller circle, then we can create the equation:

(A^2 - B^2)π = area of shaded region

Area of the smaller circle = πB^2; thus:

(A^2 - B^2)π = 3πB^2

A^2 - B^2 = 3B^2

A^2 = 4B^2

A = 2B

Since the radius of the larger circle can be expressed as 2B, the circumference of the larger circle is 4Bπ, and the circumference of the smaller circle is 2Bπ, so the circumference of the larger circle is twice that of the smaller circle.

Answer: C
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Re: In the figure shown, if the area of the shaded region is 3 t &nbs [#permalink] 01 Mar 2018, 17:35
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