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

I'm pretty sure I had the same answer, but I was trying to work it out analytically.

Undisciplined

The angle of the "pizza slice" that the green region covers is:

 and the "height" of the green region is 

science

[Daily puzzle] Find the radius of the circle

south_korea_ln

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 $$r = 6.3587$$





