- #1

skrat

- 748

- 8

## Homework Statement

Imagine having a rectangle with horizontal sides ##x## and vertical sides ##y##. The surface of that rectangle is than ##S=xy##.

Now imagine the sides ##y## are rigid - they don't bend nor do they change their length, while the ##x## sides is some kind of cloth and is easily bendable but does not extend (it does not stretch, therefore the length is always ##x##). Now if we for some reason change the distance between the parallel ##y## sides from ##x## to ##x-\delta x## the cloth will bend outwards.

The question is:

What is the shape of the cloth if we want the surface to be maximum?

## Homework Equations

## The Attempt at a Solution

Since I probably wasn't clear enough in the problem statement, here is a rough sketch (I have drawn only the upper membrane. There should be the one symmetrical at the bottom of course).

So the arc length is ##x## and the question is what is the shape of the arc? Is it a circle, an ellipse, a sin function... Whatever.

I started this way:

The total surface is $$S=(x-\delta x)y+2S_0$$ where ##S_0## is the surface under the cloth. If i am not mistaken it can be written as $$S_0=2\int_{-\Delta}^{\Delta} f(t)dt$$ where ##\Delta =\frac{1}{2}(x-\delta x)## and ##f(t)## being the function I am looking for.

I tried to solve the problem using Lagrangian multiplier and setting a constraint of constant perimeter ##2x+2y=const.##

But I didn't come very far... Is there another way? :/