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Wizard
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Wizard
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USpapergames
June 23rd, 2020 at 4:53:28 PM permalink
This question got asked on my Live Stream. I didn't want to slow down the show by working it out, so promised I would post it here after I worked it out, which I did.

Question: What is the area of a rectangle randomly inscribed in a circle of radius 10?

It may depend on how one randomly inscribes the rectangle. The way I did it was to assume a first corner of the rectangle at a specific location on the circle, say 12:00. Then I randomly chose an adjacent corner on the rectangle anywhere along the circumference of the circle. You can then draw out the rest of the rectangle based on those two points.

I'll make this one beer worthy, but usual rules apply, namely members of the beer club should not post for 24 hours. You may PM me if you can't hold your breath that long. As always, please put answers and solutions in spoiler tags until I call the problem solved.
It's not whether you win or lose; it's whether or not you had a good bet.
USpapergames
USpapergames 
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June 23rd, 2020 at 6:16:19 PM permalink
Sorry
Last edited by: USpapergames on Jun 23, 2020
USpapergames
USpapergames 
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June 23rd, 2020 at 6:18:47 PM permalink
for spoiling
Last edited by: USpapergames on Jun 23, 2020
USpapergames
USpapergames 
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June 23rd, 2020 at 6:31:28 PM permalink
the solution
Last edited by: USpapergames on Jun 23, 2020
USpapergames
USpapergames 
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June 23rd, 2020 at 6:36:07 PM permalink
I just wanted to win the reward was all :(
Last edited by: USpapergames on Jun 23, 2020
USpapergames
USpapergames 
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June 23rd, 2020 at 6:39:29 PM permalink
Your right about being able to put the rectangle at 12 o'clock but in my mind, that's still impossible since 12 o'clock is obviously straight down the middle of the rectangle since the edges should align horizontally, as if to simulate the effects of gravity and avoid edge deformation from physics forces. Btw that's just an opinion is all ;)
Last edited by: USpapergames on Jun 23, 2020
USpapergames
USpapergames 
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June 23rd, 2020 at 6:42:00 PM permalink
Btw it was me, James Jacobs who actually thought up this unique question that I've never seen in any math textbook ;)
USpapergames
USpapergames 
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June 23rd, 2020 at 6:44:10 PM permalink
Can you solve for the average rectangular prism inscribed in a sphere? I can!
USpapergames
USpapergames 
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June 23rd, 2020 at 6:45:52 PM permalink
Remember, all corners of the rectangle must touch the circumstance of the sphere & not protrude beyond the edge.
USpapergames
USpapergames 
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June 23rd, 2020 at 6:55:44 PM permalink
Here is a video of the math problem.

youtube/UpQsFChVres

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