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Inside the unique square PQRS are drawn the 4 quarter-circles having unit radii and the vertices as centers. Determine the area A that is common to the 4 quadrants.
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The area of the square is
The area of each quadrant is
Now let T be the instersection of the quarter-circle with centre Q and the quarter-circle with centre R. Then the area of the equilateral triangle TQR is
. The area of the sector bounded by RQ, RT and the arc QTS (call it ) is . This is also the area bounded by QR, QT and the arc RTP (call it ). HenceThus we have three simulataneous equations in three unknowns. So, solve them! Whee!
Last edited by JaneFairfax (2007-12-01 08:32:55)
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I got
If you wanna check, here are my answers for B and C as well.
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hi JaneFairFax
i amnot understanding how we can conclude that .
Last edited by gyanshrestha (2007-12-02 23:38:40)
http://gyan.talkacademy.com.np
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The triangle RQT is equilateral, because each of its sides is a radius.
Therefore, each of its angles is 60°.
That means that the sector A_1 has an angle of 60°, and so its area is 1/6 of the whole circle's area, which is π/6.
Why did the vector cross the road?
It wanted to be normal.
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hi JaneFairFax
i amnot understanding how we can conclude that .
Please can u give me its reasons.
A sector formed in a circle of radius r by two radii with angle θ between them has area
. (Note that θ must be in radians in the formula.)Offline
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