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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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11 Nov 2010, 08:09
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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. $$\sqrt{2}$$

[Reveal] Spoiler:
Attachment:

1111220830.jpg [ 344.39 KiB | Viewed 11934 times ]
Attachment:

Untitled2.png [ 13.24 KiB | Viewed 15807 times ]
[Reveal] Spoiler: OA

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

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

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11 Nov 2010, 08:27
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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06 Mar 2014, 01:27
Bumping for review and further discussion.

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

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09 Mar 2014, 03:10
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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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13 Apr 2014, 08:24
Bunuel wrote:

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

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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13 Apr 2014, 08:37
Bunuel wrote:

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

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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06 Dec 2017, 11:09
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. $$\sqrt{2}$$
[Reveal] Spoiler:
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.

Numbers and algebra

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

Small circle's area = x
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) = $$(4*\pi) = 4\pi$$

$$4\pi =\pi R^2$$
$$R = 2$$

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.

**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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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 write this like: πR^2=3*πr^2 (as is say 3 times)

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

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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 write this like: πR^2=3*πr^2 (as is say 3 times)

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] 25 Dec 2017, 02:24
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