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567 sats \ 5 replies \ @SimpleStacker 11 Oct 2024 \ on: [Daily puzzle] Find the radius of the circle science
Ok, for real this time. Let \ell be the length of the rectangle.
We know r\ell = 0.5\pi r^2 and so \ell = 0.5\pi r
Moreover, it'll be useful to write: \ell - r = (0.5\pi - 1)r
The angle of the "pizza slice" that the green region covers is:
\theta = \cos^{-1} \left( \frac{ \ell - r }{r} \right) = \cos^{-1} (0.5\pi -1)
The area of the pizza slice is 0.5\theta r^2 and the "height" of the green region is r \sin \theta .
Thus:
0.5\theta r^2 - 0.5(\ell - r)r\sin \theta = 10
Or:
\left[ 0.5 \theta - 0.5(0.5\pi - 1)\sin \theta \right] r^2 = 10
Which gives r = 6.3587
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20 sats \ 4 replies \ @Undisciplined 11 Oct 2024
I'm pretty sure I had the same answer, but I was trying to work it out analytically.
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21 sats \ 3 replies \ @SimpleStacker 11 Oct 2024
I'm not as much of a fan of these brute force calculations, but once I start on something I find it very hard to stop...
I gave myself 15 minutes to do this, and I thought I had it, but it turns out I made a mistake. Ended up spending maybe 30-40 minutes on this problem... not great for my productivity today lol
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0 sats \ 2 replies \ @Undisciplined 11 Oct 2024
I don't think it simplified nicely, which isn't surprising when you put stuff like that into an inverse cosine.
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39 sats \ 1 reply \ @south_korea_ln OP 12 Oct 2024
Yeah, i also usually prefer the ones that simplify nicely. Also because it's an indirect confirmation one is on the right track.
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94 sats \ 0 replies \ @Undisciplined 12 Oct 2024
It feels like a better puzzle when the answer comes out nice and simple.
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