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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
Attachment:
2017-06-19_1058.png [ 10.07 KiB | Viewed 12075 times ]
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19 Jun 2017, 00:16
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Bunuel
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
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)...
Updated on: 19 Jun 2017, 14:09
Originally posted by quantumliner on 19 Jun 2017, 00:24.
Last edited by quantumliner on 19 Jun 2017, 14:09, edited 1 time in total.
Last edited by quantumliner on 19 Jun 2017, 14:09, edited 1 time in total.
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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
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
