# Surface w/ max volume and min surface area

1. May 16, 2006

### ksle82

I'm attempting to solving this problem but do not know how to begin. Any help would be appreciate.

What geometric surface encloses the maximum volume with the minimum surface area? How would you prove it?

2. May 17, 2006

### arildno

Well, first of all, you should state your question more precisely:
Alternative 1:
Of all geometric objects with the same volume, which has minimal surface area?

Alternative 2:
Of all geometric objects with the same surface area, which encloses the maximal volume?

Under certain assumptions of niceness, you may solve problems like these with the calculus of variations.

Yhen, in both cases, the ball (solid sphere) will be your solution.

3. May 17, 2006

### J77

If you have some constraints, use Lagrange multipliers.

4. May 17, 2006

### rcgldr

A soap bubble provides an excellent model. The surface tension in a bubble causes the surface area to be minimized. Since it forms a sphere, this is evidence that the minimal surface area for a set volume of air is a sphere.

5. May 17, 2006

### jasc15

Since there is only one answer to this question (i.e., a sphere) is there a proof for this? Intuitively it makes sense that it would be a sphere, but there must be a rigorous way to demonstrate this..

6. May 17, 2006

### Curious3141

I'm out of touch with it, but I'm pretty sure you can use the Calculus of Variations to do it.

7. May 17, 2006

### Tantoblin

Isn't this related to the (very hard) Plateau problem?

8. May 17, 2006

### HallsofIvy

Yes, the general study of functions giving maximal values (as opposed to numbers) is precisely the "Calculus of Variations". And, yes, one can prove, using the calculus of variations, that, under certain conditions, the surface enclosing maximum volume for given surface area, or, conversely, having minimum surface area for given volume, is a sphere.
There do, however, exist rather peculiar 'special surfaces' that also have those properties.

9. May 17, 2006

### ksle82

Well, I have a feeling that the answer would be a sphere but I just don't know how to solve it using calculus. Any hint would help alot.

10. May 17, 2006

### rcgldr

What would these surfaces look like? Could they happen in the real world, for example, would a soap bubble every take on one of these special shapes?

11. May 18, 2006

### arildno

Typically, those surfaces that cannot be discerned by standard variational techniques, yet represent extremizing values, are non-smooth surfaces.

12. May 18, 2006

### fourier jr

this is one of the isoperimetric problems proposed by the ancient greeks; one was to find the plane figure with maximal area if given a prescribed perimeter. the other was the other way around, to find the plane figure with minumum perimeter given a prescribed area. the answer is a circle (or sphere i 3 dimensions i guess); jacob steiner proved it in the 1700s or whenever he was around.

13. Jul 7, 2006

### vishak31102005

proof of calculation

hi guys

would it possible for somebody to actually write out the mathematical calculations for the calculation of miminum surface area of a bubble using the Euler-Lagrange Differential Equation.

thanks
vishak

14. Jul 7, 2006

### Robokapp

Well you just know...I guess if you try to make a garden you don't want to make it very long and very naroow or you'll lose surface. A cubic box holds more than a really wide, really low one. I mean reducing to absurd, a box with width and length of 10 and height of 1 has a volume of 100 and one with 33 on each side has a volume of 33^3. A tethrahedral box takes far less than a cubic one...and so on.