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18 Aug 2021, 08:00
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C
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Difficulty:
65%
(hard)
Question Stats:
52% (01:40) correct
48%
(02:01)
wrong
based on 240
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In the image above, two tangent semicircles are drawn in a circle of area 4. If the diameters of the two semicircles are parallel, what is the area of the shaded region?
A. 1
B. 1.5
C. 2
D. 2.5
E. 3
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M37-37
Attachment:
Untitled4.png
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23 Nov 2021, 07:29
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Official Solution:
In the image above, two tangent semicircles are drawn in a circle of area 4. If the diameters of the two semicircles are parallel, what is the area of the shaded region?
A. \(1\)
B. \(1.5\)
C. \(2\)
D. \(2.5\)
E. \(3\)
The question has five exact answer choices (no "Cannot be determined" or "None of the above") so one of them must be correct. If one of the answers is correct, the answer must be correct no matter how we draw the diagram (of course we should not violate info that is given).
Re-draw the diagram so that the upper semicircle is infinitely small. In this case the bottom semicircle will occupy half of the circle, and thus the area of the shaded region (which will be just one semicircle now) would be half of the larger circle, so 2.
Answer: C
In the image above, two tangent semicircles are drawn in a circle of area 4. If the diameters of the two semicircles are parallel, what is the area of the shaded region?
A. \(1\)
B. \(1.5\)
C. \(2\)
D. \(2.5\)
E. \(3\)
The question has five exact answer choices (no "Cannot be determined" or "None of the above") so one of them must be correct. If one of the answers is correct, the answer must be correct no matter how we draw the diagram (of course we should not violate info that is given).
Re-draw the diagram so that the upper semicircle is infinitely small. In this case the bottom semicircle will occupy half of the circle, and thus the area of the shaded region (which will be just one semicircle now) would be half of the larger circle, so 2.
Answer: C
General Discussion
18 Aug 2021, 08:42
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As the question doesn't say anything about the radius of the internal circles we can suppose that they have the same radius. If so, the tangent point between them is in the center of the bigger circle.
We can picture a right triangle with that center, the center of a semicircle and the point where the semicircle touch the bigger circle.
Let:
R: radius bigger circle
r: radius semicircle
We have:
\(pi*R^2=4\) -> \(R^2=\frac{4}{pi}\)
\(R^2=r^2+r^2=2r^2\) -> \(r^2=\frac{R^2}{2}=\frac{2}{pi}\)
The shaded region is:
\(pi*r^2=pi*\frac{2}{pi}=2\)
IMO C
We can picture a right triangle with that center, the center of a semicircle and the point where the semicircle touch the bigger circle.
Let:
R: radius bigger circle
r: radius semicircle
We have:
\(pi*R^2=4\) -> \(R^2=\frac{4}{pi}\)
\(R^2=r^2+r^2=2r^2\) -> \(r^2=\frac{R^2}{2}=\frac{2}{pi}\)
The shaded region is:
\(pi*r^2=pi*\frac{2}{pi}=2\)
IMO C
