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Shifted regions (Posted on 2019-02-27) Difficulty: 2 of 5
A circle is centered at the origin with a radius of 5. Two lines are drawn that divide the circle into four regions, the first is the line y=-1, and the other is x=-3. Find the area of the largest region.

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Solution solution | Comment 2 of 4 |
The largest region is of course the one containing the center/origin. 

The arc measure in radians of this region is pi/2 + arcsin(3/5) + arcsin(1/5). The portion which is a sector of the circle therefore has area 25*(pi/2 + arcsin(3/5) + arcsin(1/5))/2.

There is also a right triangle with legs of lengths 1 and 5 and another with legs of lengths 3 and 5 as well as a 3x1 rectangle, adding 5/2 + 15/2 + 3 = 13 to the total.

The largest region therefore has area 25*(pi/2 + arcsin(3/5) + arcsin(1/5))/2 + 13 ~= 43.1956919547313

Correction, upon examining my work in regard to Steven Lord's comment:

The second right triangle does not have leg lengths of 3 and 5, but rather 3 and 4 (it's the hypotenuse that is length 5), to its area is 6, rather than 15/2. The area should be

25*(pi/2 + arcsin(3/5) + arcsin(1/5))/2 + 5/2 + 6 + 3 ~= 41.6956919547313

And now I see another error. The smaller right triangle's longer leg is not 5, but rather cos(arcsin(1/5)).


25*(pi/2 + arcsin(3/5) + arcsin(1/5))/2 + 5*cos(arcsin(1/5))/2 + 6 + 3 ~= 41.6451816975145

Edited on February 28, 2019, 3:33 pm
  Posted by Charlie on 2019-02-27 15:09:30

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