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[Anonymous No.11483443]

A circular disk of paper is cut into two sectors with angles θ and 2π-θ. The two sectors are perfectly rolled up into two cones which are taped closed with no paper overlapping. What angle should the cut be to maximise the combined volumes of the two cones?

[Anonymous No.11483446]

>>11483443
Using Wolfram is allowed by the way

[Anonymous No.11483451]

pi

[Anonymous No.11483453]

2 pi + 2 thet@

[Anonymous No.11483633]

>>11483446
If can't solve that without mathematica you should stop posting here
>>11483451
Agree

[Anonymous No.11483678]

3/5 pi

[Anonymous No.11483723]

>>11483451
>>11483453
>>11483678
Wrong. It's

1/2 (2 π - sqrt(4 π^2 - 4 (-(20 π^2)/9 + (86 2^(2/3) (1 + i sqrt(3)) π^2)/(9 (1121 + 9 i sqrt(191))^(1/3)) + 1/9 (1 - i sqrt(3)) (2 (1121 + 9 i sqrt(191)))^(1/3) π^2)))

[Anonymous No.11483727]

>>11483723
Are you retarded? the only way to prove you're correct is by actually building the cones.

[Anonymous No.11483733]

>>11483727
Or you could just use high school level calculus.

[Anonymous No.11483741]

>What angle should the cut be to maximise the combined volumes of the two cones?
*Maximize

Either way this question is retarded because both cones will add up to the same volume.

[Anonymous No.11483743]

>>11483723
...which is pi

[Anonymous No.11483750]

>>11483743
It's not pi

[Anonymous No.11483755]

I need to solve a really high order equation to get the answer.

[Anonymous No.11483948]

>>11483755
Could you explain?

[Anonymous No.11483969]

Trick question, the cones will be open and thus not have a volume

[Anonymous No.11484003]

>>11483755
>>11483948

The volume is (1/3)(pi)(R^2)h, but 1/3 and pi can be ignored. R is theta/2pi. h is sqrt((the radius of the circle, or 1)^2 - (the cone radius, or (theta/2pi))^2 aka sqrt((r^2)-(theta/2pi)^2). Replace r with 1, and theta with x and add a copy the expression with 2pi-x instead of x and I got the following in desmos.

\left(\frac{x}{2\pi}\right)^{2}\sqrt{1^{2}-\left(\frac{x}{2\pi}\right)^{2}}+\left(\frac{\left(2\pi-x\right)}{2\pi}\right)^{2}\sqrt{1^{2}-\left(\frac{\left(2\pi-x\right)}{2\pi}\right)}^{2}

This is obviously wrong as it isn't symmetrical, but idk what I did wrong.

[Anonymous No.11484154]

>>11483443
>What angle should the cut be to maximise the combined volumes of the two cones?
Trick question. The volume would be the same regardless of the angle of the cut.

[Anonymous No.11484175]

>>11483741
>>11484154
Holy shit brainlets

[Anonymous No.11484732]

>>11484175
its correct, its just not the answer OP wants since they did not explicitly state the volume inside the cone must be considered....

[Anonymous No.11484754]

>>11483443
Okay, but answer me this first. What is a sector?

[Anonymous No.11485150]

>>11484754
Brainlet OP whats to cut two pieces of a circular paper to make two cones. His brainlet question is how large should the angles be to maximize the sum of the volume of the cones.
>Pic related

>Brainlet op doesn't understand that the volume will add up to the same number regardless of what the angle is.

[Anonymous No.11486033]

>>11485150
Incorrect, try again.

[Anonymous No.11486138]

>>11485150
If I understand it correctly, we have 2 disks with equal radius and we cut both at the same angle.

We can reduce that problem to maximizing the volume of a cone with set slant height and with respect to the pointy angle at the top.

[Anonymous No.11486168]

>>11486138
You have one disk of radius 1 and you pick two radii on the disk to cut it into two pieces, and each piece makes one cone with side length 1 and circumference equal to either θ or 2π-θ, depending on which piece it is.

[Anonymous No.11486211]

>>11486168
that makes 2 cones with different radii in the base

[Anonymous No.11486223]

>>11483443
[math]\frac{\pi}{2}[/math]

[Anonymous No.11486267]

>>11486211
Yes, you want to maximise the combined volume of cone1+cone2, not one of them specifically.

[Anonymous No.11486369]

>pi/2

fucking brainlets

[Anonymous No.11486410]

>>11486369
I don't know what you've done but it's clearly wrong. If you have two cones with constant slope length and circumference θ and 2π-θ then the volume should only approach 0 as one of θ and 2π-θ approach 0.

[Anonymous No.11486688]

>>11486410

tfw can't read scale or basic math

[Anonymous No.11486727]

>>11486369
Lmao

[Anonymous No.11486738]

>>11486410
The two cones don't have constant slope or length.

[Anonymous No.11486765]

>>11486738

Anon is wrong but he meant slope length as in radius of the disk that gets cut up which is always the same

[Anonymous No.11486767]

>>11483723
Haha, what the fuck is all that shit? The fucking nerds on this board, I swear

[Anonymous No.11486876]

>>11486033
You have to be retarded or something

[Anonymous No.11486891]

>>11483723
How do you calculate this?

[Anonymous No.11487025]

I solved this a couple days ago. If you let pi-x and pi+x be the two angles, you get symmetry that makes it solvable. After taking the derivative and setting it to zero, square things to get rid of the square roots. You are left with a seventh degree polynomial with no constant term. Divide out x, you get a 6th degree polynomial with only even-powered terms. Substitute z = x^2 to get a cubic. Solve that to get z, then get x.

You can avoid complex numbers if you use the trigonometric solution of the cubic.

[Anonymous No.11487081]

>>11486891
A good building should not show its scaffolding when completed.

[Anonymous No.11487088]

>>11486876
Incorrect, try again. Hint: your answer should be a number.

[Anonymous No.11487104]

>>11483443
>>11487088
pi/2

[Anonymous No.11487112]

>>11487104
Wrong, try again.

[Anonymous No.11487362]

>>11486369
Yes, I misread and thought θ and 2π-θ was the angles of the cuts, giving sectors with angles 2θ and 2π-2θ. I think your variable rp is wrong.

[Anonymous No.11487371]

>>11483741
*Maximise

[Anonymous No.11487426]

>>11487362

Naw ur just retarded

With θ' = τ - θ and R = disk radius:
Radius of sector with angle θ:
(τR)(θ/τ)(1/τ) = θR/τ

Radius of other sector:
(τR)((τ-θ)/τ)(1/τ) = R(τ-θ)/τ = Rτ/τ - Rθ/τ = R - r

[Anonymous No.11487561]

>>11487025
based

[Anonymous No.11487578]

>>11487426

[Anonymous No.11487826]

>>11487578
You're confusing the radius of the disk with the radius of the cone.

[Anonymous No.11487828]

>>11487578
The radius isn't the same for both cones