[Daily puzzle] Find the radius of the circle \ stacker news ~science

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[Daily puzzle] Find the radius of the circle

1926 sats \ 26 comments \ @south_korea_ln 11 Oct science

In the same vein as yesterday's problem, find the radius $r$ such that the area of the semicircle is the same as the area of the rectangle. The green region has an area of 10 (a.u.).

Previous iteration: #717118

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16 sats \ 0 replies \ @Undisciplined 11 Oct

Should there be an elegant solution or is it a big mess of radicals and inverse trig functions?

567 sats \ 5 replies \ @SimpleStacker 11 Oct

Ok, for real this time. Let

be the length of the rectangle.

We know

and so

Moreover, it'll be useful to write:

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

The area of the pizza slice is

and the "height" of the green region is

Thus:

Or:

Which gives

20 sats \ 4 replies \ @Undisciplined 11 Oct

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

21 sats \ 3 replies \ @SimpleStacker 11 Oct

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

0 sats \ 2 replies \ @Undisciplined 11 Oct

I don't think it simplified nicely, which isn't surprising when you put stuff like that into an inverse cosine.

39 sats \ 1 reply \ @south_korea_ln OP 12 Oct

Yeah, i also usually prefer the ones that simplify nicely. Also because it's an indirect confirmation one is on the right track.

94 sats \ 0 replies \ @Undisciplined 12 Oct

It feels like a better puzzle when the answer comes out nice and simple.

521 sats \ 5 replies \ @Scroogey 11 Oct

The green area is half a circle segment with height h.

The area of the rectangle is r*(r + r - h).

The area of the semicircle is (pi * r^2) / 2.

Since they are equal, you get h = 2*r - (pi * r)/2

Insert that h into the circular segment area based on height h formula and you get

r ~ 6.3587

22 sats \ 3 replies \ @zapsammy 11 Oct

the reason this would take a while for me is that i like to derive the formulas from scratch for such problems... otherwise i trust not verify, haha.

43 sats \ 2 replies \ @SimpleStacker 11 Oct

sometimes being good at math is knowing that someone else derived the formula you need, and knowing where to find that formula.

1 sat \ 1 reply \ @zapsammy 11 Oct

yes, it's a process... i have the stamina and interest to relearn geometry, because i have a dream to build in stone, make true masonry great again.

i actually ran into a real mason recently. he retired and started making canvas art instead, based on his work as a mason. very intricate, never seen anything like it.

43 sats \ 0 replies \ @SimpleStacker 11 Oct

That's a very cool reason to want to learn geometry. I wish you luck in your pursuits!

1 sat \ 0 replies \ @south_korea_ln OP 12 Oct

Nice find, didn't even though there was a formula available for this on Wikipedia

32 sats \ 7 replies \ @0xbitcoiner 11 Oct

This is another level. I'm not even going to try!

11 sats \ 6 replies \ @south_korea_ln OP 11 Oct

11 sats \ 5 replies \ @0xbitcoiner 11 Oct

I'm just not going to try it because I've got a headache and I don't want to make it worse! Today is Friday ... Ahahaha

42 sats \ 4 replies \ @south_korea_ln OP 11 Oct

We say 불금 here, fire-friday... people getting drunk and letting free after a hard week of work. No alcohol for me today, though...

0 sats \ 3 replies \ @0xbitcoiner 11 Oct

this? 🤠

I don't need a lot of alcohol to have fun. Two whiskies is enough for the whole night! Ahhahaha

42 sats \ 2 replies \ @south_korea_ln OP 11 Oct

Social media campaign from Hana Bank it seems.

A modern take on 불금 where one prefers staying in with some fried chicken and beer (chimaek, 치맥) to watch a Korean drama on the laptop~~

11 sats \ 1 reply \ @0xbitcoiner 11 Oct

Ah ok! Or read Han Kang's works. #717216

21 sats \ 0 replies \ @south_korea_ln OP 11 Oct

Missed that post. Tnx for linking. Enjoy, time for me to sleep here.

reply on another page

0 sats \ 0 replies \ @zapsammy 11 Oct

i have a bet with an older friend that i will pass the differential calculus exam when i reach the age at which he failed it (about 40yo), after having the same amount of study time. i shall also pass the integral calculus exam, because it's not about the age, but the amount of cobwebs in the brain.

0 sats \ 0 replies \ @WeAreAllSatoshi 11 Oct

We should have a button for MathJax syntax like the markdown instructions button in the upper right of the text area input. I am familiar with markdown, but not MathJax.