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# Area of the overlap

Best Answer superprismatic, 18 February 2013 - 06:24 PM

Spoiler for I believe it is

Go to the full post

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Posted 18 February 2013 - 04:22 AM

Suppose you have a unit circle. You cut the circle into four equal parts. Where the radii of one of the four pieces exists (0,0) to (0,1) and (0,0) to (1,0). Putting one of the four pieces on each of the following coordinates, (0,0) , (0,1) , (1,1) , (1,0) and fitting them together so that the radii produce a square causes the interior to have overlapping arcs. There is a piece, in the innermost center, where all four arcs overlap. What is the area of the overlapped center?
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### #2 superprismatic

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Posted 18 February 2013 - 06:24 PM   Best Answer

Spoiler for I believe it is

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### #3 CaptainEd

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Posted 20 February 2013 - 07:20 PM

Spoiler for geometry solution

Edit: Sorry folks, I don't know why my spoilers didn't hide... :-(

Edit: correct sign on sqrt(3)

Edited by CaptainEd, 20 February 2013 - 07:27 PM.

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### #4 Yoruichi-san

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Posted 20 February 2013 - 08:09 PM

Lol nice.  Hooray for calculus ;P.  I had to explain a similar problem to a kid I was tutoring for SAT II Math and I couldn't use calc...it was a pain (the explaining, not the kid).

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### #5 bonanova

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Posted 20 February 2013 - 08:22 PM

I learned calculus once. I remember using the word integral correctly in a sentence.

For most of my career I used algebra and computers to solve problems. Since retirement,

I differentiate, but only in the logical sense.

I am in awe that SP did what he did; I understand what the Cap'n did.

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### #6 CaptainEd

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Posted 20 February 2013 - 09:03 PM

For that matter,  I'd like to know more about how SP did what he did. My calculus, too, is out in the shed, gathering rust, plant food, and spider webs. But *maybe*  I could follow it.

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### #7 superprismatic

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Posted 20 February 2013 - 10:37 PM

Spoiler for You asked for it, so.....

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### #8 Yoruichi-san

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Posted 20 February 2013 - 11:04 PM

Yeah the integral is pretty ugly...I think you have to do it by parts and the second part will end up being a hyperbolic sine :/.

Coming up with the integral is the fun part, IMO...in my college days we had a school-wide license for Mathematica ;P.

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Women are definitely stronger. We are [Fe]males, after all...

Some of what makes me me is real, some of what makes me me is imaginary...I guess I'm just complex. ;P

<3 BBC's Sherlock, the series and the man. "Smart is the new sexy."

Chromatic Witch links now on my 'About Me' page! Episode 3 is finally here!

When life hands me lemons, I make invisible ink.

### #9 CaptainEd

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Posted 20 February 2013 - 11:31 PM

Thank you, SP and Y-san, I remember enough about calculus to know what you're saying, and even remember the notion of integration by parts. And I've heard of Mathematica. I can spell wxMaxima. I'll bet it took me more hours to do the geometry than it took you to do the calculus. (My geometry is stored in the mudroom right outside the shed)