Earlier I devised a deceptively simple-seeming maths problem. To my annoyance I can't solve it:
You want to cut a circular cake, diameter d, into x pieces of equal area.
The traditional way of doing this would be to cut normal slices every 360/x deg around the circle.
However you are an autist from 4chan and need each slice to have the exact same area - you also only have a ruler to hand.
Therefore devise an equation/ set of equations to cut the cake in parallel slices into slices of exactly equal area simply in terms of x and d.
Please note, this is NOT a homework request and I will be pissed off if anybody suggests it is, I devised this problem and I think it leads to a procedure not an equation but am really not sure - if you can begin to understand it you will hopefully see that it is at a level of maths higher than one at which you might be given "homework", thank you
numberphile answered this question
based numberphile
>>8082099
You have misunderstood the word "parallel" I fear
>>8082083
https://en.wikipedia.org/wiki/Circular_segment#Area
doesn't look pretty
Would you mean something like this?
>>8082188
It's cool how the image compression on the thumbnail makes the perfectly straight lines appear wobbley and bent.
>>8082083
1/2
>>8082083
2/2
>>8082083
3/2:
this is the best i could come up with. you can reduce the problem to a recursion relation subject to the condition that you start at the origin and build up outwards to a_x = R. each a_n gives you a little slice, but there isn't any regular pattern to them, and i definitely dont think that the recursion relation can be summed into something expressible in terms of elementary functions. solving for the first width is impossible analytically bc it's a transcendental equation from the get go. maybe there's a totally different approach that's tractable, though.
>>8082083
>level of maths
plug chug isn't any level of maths
you don't even have to think at all, an expression for the answer is obtained directly.
>>8082083
What do you mean by the area of the piece? The surface area? The area of a face? Or do you actually mean the volume?
>>8082083
you need height of cake for thin cakes
>>8082121
The area of the circle is A=d^2*pi/4. You want x>1 slices, so you want to make cuts which would cut off circular segments of areas
A=y*d^2*pi/(4 x), for y=1,2,...,x-1. The circular segement area formula is A = d^2/8 (T-sin(T)), where T is the central angle in radians, so
2*y*pi/x = T-sin(T). Assume the cake is centered at the origin and we make cuts parallel to the horizontal axis. The cut is then at the
location c=R*sin(T/2) up the vertical axis. Then
2*asin(c/R)=T, so 2*y*pi/x = 2*asin(c/R) - sin( 2*asin(c/R) ). We need to solve this for c. Now sin( 2*asin(c/R) simplifies to (2 c sqrt(1-c^2/R^2))/R, but solving explicitly for c in
2*y*pi/x = 2*asin(c/R) - (2 c sqrt(1-c^2/R^2))/R
looks too difficult.
>>8082954
It's not solvable explicity.
>>8082083
It's very simple but can only be solved numerically.
y - sin y = 2 F pi
Where F is the fraction of the area a particular chord divides the circle into and y is the angle between the radii which connect the endpoints of that chord (this interior angle defines the chord). So the first chord is
y - sin y = 2 pi / x
The second is
y - sin y = 4 pi / x
Etc.
