How should the radius MP be drawn?

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The discussion focuses on determining how to draw the radius MP in a geometric configuration defined by three semicircular arches, where the smallest arch has a radius denoted as r. The objective is to divide the colored area into two equal parts by finding the optimal position of point P. The proposed method involves quantifying the position of P, calculating the areas of the left and right segments of the blue region, and solving for P without the need for integration. The left area is calculated as one-fourth of the large circle's area plus half of the small circle's area.

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I need help with this one.

http://g.imagehost.org/view/0971/24_p_181


The coloured parts space is limited by three semicircle archs of which the smaller has the radius r.

How should the radius MP be drawn, if you want it to divide the coloured space into two parts with
equally big areas?
 
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What have you tried? Where are you stuck? At first glance, it would look like the most obvious thing would work:

1. Find a convenient way to quantify the position of P
2. Compute a formula for the area of the left blue part
3. Compute a formula for the area of the right blue part
4. Solve for the position of P that makes those areas equal
 
This doesn't require any integration at all. If you draw a vertical downward from M, you cut the blue region into two parts. The area of the left part is 1/4 the area of the large circle plus 1/2 the area of the small circle. What is that?
 

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