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In the figure to the right, the radius of the larger circle is twice

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In the figure to the right, the radius of the larger circle is twice [#permalink]

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New post 19 Jun 2017, 00:02
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In the figure to the right, the radius of the larger circle is twice that of the smaller circle. If the circles are concentric, what is the ratio of the shaded region’s area to the area of the smaller circle?

(A) 10:1
(B) 9:1
(C) 3:1
(D) 2:1
(E) 1:1


[Reveal] Spoiler:
Attachment:
2017-06-19_1058.png
2017-06-19_1058.png [ 10.07 KiB | Viewed 1021 times ]
[Reveal] Spoiler: OA

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In the figure to the right, the radius of the larger circle is twice [#permalink]

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New post 19 Jun 2017, 00:16
Bunuel wrote:
Image
In the figure to the right, the radius of the larger circle is twice that of the smaller circle. If the circles are concentric, what is the ratio of the shaded region’s area to the area of the smaller circle?

(A) 10:1
(B) 9:1
(C) 3:1
(D) 2:1
(E) 1:1


[Reveal] Spoiler:
Attachment:
2017-06-19_1058.png


Let the Radius of Larger Circle be \(= R\)
Area of Larger Circle = \(\pi\)\(R^2\)

Let the Radius of Smaller Circle be \(= r\)
Area of Smaller Circle = \(\pi\)\(r^2\)

Given radius of larger circle is twice that of smaller circle. ie; \(R = 2r\)
Therefore Area of Larger Circle = \(\pi\)\({(2r)}^2\) = \(\pi\)4\(r^2\)

Area of Shaded region = Area of Larger Circle - Area of Smaller Circle
= \(\pi\)4\(r^2\) - \(\pi\)\(r^2\)
= \(\pi\)\(r^2 (4 - 1)\)
= \(\pi\)\(r^2 (3)\)
Area of Shaded region \(= 3\)\(\pi\)\(r^2\)

Ratio of the shaded region’s area to the area of the smaller circle = Area of Shaded region / Area of Smaller circle

Required ratio \(= 3\)\(\pi\)\(r^2\) / \(\pi\)\(r^2\) \(= \frac{3}{1}\) or \(3 : 1\). Answer (C)...

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In the figure to the right, the radius of the larger circle is twice [#permalink]

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New post 19 Jun 2017, 00:24
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Radius of small circle = r , Area of small circle =\(\pi\)\(r^2\)
Radius of large circle = 2r,Area of large circle =4\(\pi\)\(r^2\)

Area of shaded region = Area of large circle - Area of small circle = 4\(\pi\)\(r^2\) - \(\pi\)\(r^2\) = 3\(\pi\)\(r^2\)

Ratio of Area of shaded region to Area of small circle = 3\(\pi\)\(r^2\) / \(\pi\)\(r^2\) = 3

Answer is (C) 3:1

Last edited by quantumliner on 19 Jun 2017, 14:09, edited 1 time in total.

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Re: In the figure to the right, the radius of the larger circle is twice [#permalink]

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New post 19 Jun 2017, 04:59
Area of Circle = \(2 \pi r\)
Required Ratio = \(\frac{Area of Shaded Region}{Area of Smaller Circle}\)
----- eq 1

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

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

Given that, R = 2r

Substituting values in eq 1

\(\frac{(2 \pi (2r)^{2} - 2 \pi r^2)}{2 \pi r^2}\)

\(\frac{2 \pi r^2 (4-1)}{2 \pi r^2}\)

\(\frac{3}{1}\)

Ratio = 3: 1

Ans : Option C

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Re: In the figure to the right, the radius of the larger circle is twice [#permalink]

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New post 19 Jun 2017, 11:26
Let the radius of smaller circle be r. Then the radius of bigger circle will be 2r.
Area of smaller circle will be πr2 and that of bigger circle will be 4πr2.
Area of shaded region will be 4πr2 - πr2 = 3πr2
Ratio of shaded region to smaller circle will be 3πr2:πr2 = 3:1
Option C is correct

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Re: In the figure to the right, the radius of the larger circle is twice [#permalink]

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New post 19 Jun 2017, 15:57
C)

Area of larger circle = pi*2r^2 = 4*pi*r^2
Area of smaller circle = pi*r^2

Area of shaded region = 4*pi*r^2 - (pi*r^2) = 3*pi*r^2

ratio = 3*pi*r^2/pi*r^2 = 3/1

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In the figure to the right, the radius of the larger circle is twice [#permalink]

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New post 19 Jun 2017, 18:10
Bunuel wrote:
Image
In the figure to the right, the radius of the larger circle is twice that of the smaller circle. If the circles are concentric, what is the ratio of the shaded region’s area to the area of the smaller circle?

(A) 10:1
(B) 9:1
(C) 3:1
(D) 2:1
(E) 1:1


[Reveal] Spoiler:
Attachment:
2017-06-19_1058.png

Let large circle radius = 4

Let small circle radius = 2

(Large circle area) - (small circle area) = (shaded region area)

Area of circle =\(\pi r^2\)

Large circle area = \(16\pi\)

Small circle area = \(4\pi\)

Shaded region area: \(16\pi\) - \(4\pi\) = \(12\pi\)

Ratio of shaded area to small circle area

\(\frac{12\pi}{4\pi}\) =

\(\frac{3}{1}\)

Answer
[Reveal] Spoiler:
C

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Re: In the figure to the right, the radius of the larger circle is twice [#permalink]

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New post 20 Jun 2017, 18:45
Bunuel wrote:
Image
In the figure to the right, the radius of the larger circle is twice that of the smaller circle. If the circles are concentric, what is the ratio of the shaded region’s area to the area of the smaller circle?

(A) 10:1
(B) 9:1
(C) 3:1
(D) 2:1
(E) 1:1



We can let the radius of the small circle = r, and thus the radius of the large circle = 2r. Let’s first determine the area of the shaded region:

area = (radius of large circle^2 - radius of small circle^2)π

area = [(2r)^2 - r^2)π

area = (4r^2 - r^2)π = (3r^2)π

We also see that the area of the smaller circle is πr^2.

Thus, the ratio of the shaded region’s area to the area of the smaller circle is:

(3r^2)π/πr^2 = 3/1 = 3 : 1

Answer: C
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Re: In the figure to the right, the radius of the larger circle is twice   [#permalink] 20 Jun 2017, 18:45
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