# Thread: Maximizing volume of an ellipsoid with a given surface area.

1. ## Maximizing volume of an ellipsoid with a given surface area.

I have a problem for my vector calculus class where we have to build a storage tank that is the shape of a surface of revolution obtained by rotating
x^2 + a^2 y^2 = 1, x > 0, around the y-axis. Given a fixed surface area S the problem is to find the best value of a to maximize the volume.

I'm honestly so fried with abstract classes that I'm having trouble approaching this problem. Any help would be greatly appreciated.

2. ## Re: Maximizing volume of an ellipsoid with a given surface area.

Originally Posted by anoldoldman
I have a problem for my vector calculus class where we have to build a storage tank that is the shape of a surface of revolution obtained by rotating
x^2 + a^2 y^2 = 1, x > 0, around the y-axis. Given a fixed surface area S the problem is to find the best value of a to maximize the volume.

I'm honestly so fried with abstract classes that I'm having trouble approaching this problem. Any help would be greatly appreciated.
The basic idea is that you use your constraint, i.e. the surface area, to reduce the expression for the volume from 2 variables to 1 variable. Via the constraint you'll have an expression for y in terms of x or vice versa, whichever is more convenient, and you plug that into your volume formula.

That whole mess gives you the volume as the function of a single variable and you know how to maximize that. Just find the zero of the derivative.

Of course you have to come up with formulas for the surface area and volume of your surface of revolution but you can figure that out.

(in case you can't remember you make a surface by spinning a curve around the center 2pi radians. To make a volume you spin the area under a curve around the center 2pi radians.
In the first case the surface area is the length of the curve times 2pi. For the volume it's the area under the curve times 2pi.)

3. ## Re: Maximizing volume of an ellipsoid with a given surface area.

I'm trying to get the arc length of the curve but it is coming out to a horrible integral. I created the integral by solving for y and then running it through the standard arc length formula. Is there a better way of doing this that I am missing?

4. ## Re: Maximizing volume of an ellipsoid with a given surface area.

Originally Posted by anoldoldman
I'm trying to get the arc length of the curve but it is coming out to a horrible integral. I created the integral by solving for y and then running it through the standard arc length formula. Is there a better way of doing this that I am missing?
Looking into this I find that calculating the arc length of a 1/4 ellipse is a not trivial. Here is a reference.

Calculating the area of a 1/4 ellipse is at least easy. It's pi/(4a) based on how you've specified your ellipse. So the full area is just pi/a.

Your problem btw is equivalent to maximizing the area of the 1/4 ellipse subject to arc length S/2pi for the same quarter ellipse since the surface areas and volumes of your revolved object is just these times 2pi.

This is equivalent to maximizing the area of the ellipse for a constrained perimeter of (2S)/pi.

Here is a page on calculating the circumference of an ellipse.

5. ## Re: Maximizing volume of an ellipsoid with a given surface area.

I've been looking at this problem some more and it's poorly posed.

You want a function x^2 + a^2 * y^2 = 1 and you want to vary only "a" but doing so will inevitably change the arc length of the quarter ellipse and thus the surface area of your surface of revolution.

You need another scaling parameter so that you can select an ellipse eccentricity via "a" and then scale the ellipse so it's arc length remains constant.

Does this make sense?