Optimization of ellipsoid tube

[synergix]

Homework Statement

Problem 2 b) in the following link

http://www.math.ubc.ca/~haber/courses/math253/Welcome_files/asgn4.pdf"

Homework Equations

V=pi(r1r2)H
SA=?

The Attempt at a Solution

I was thinking I should form two equations V=10=pi(r1r2)h and then an equation for the surface area and then optimize the two. However, the equation for circumference of an ellipse seems to be something of a troublesome thing. Should I pick an equation that I think will be the best? Such as the Hudson equation? I have never heard of the Hudson equation before but I found it here: (http://local.wasp.uwa.edu.au/~pbourke/geometry/ellipsecirc/)

Thank you for taking the time to look this over!

 

[marcusl]

Well, the assignment says to be creative. Think about fixing the height and finding the shape of the ellipse of constant area having the smallest circumference.
HINT: what happens to the circumference as the eccentricity becomes large and the ellipse flattens towards two parallel lines? What is the opposite case?

 

[synergix]

Well I know from experience that a circular cylinder will have the maximum volume. I am also pretty sure it will have the smallest circumference. If this is correct then now I must show it? At what value would I fix h? Do I just leave it as a constant and find it later after I know the optimum values of r1 relative to r2?

 

[marcusl]

Yes, I'd just leave h for later and concentrate on finding the optimal ellipse eccentricity (that is, highest ratio of area to circumference). You experience is pointing to the right answer.

 

[synergix]

So in order to do this I need to find circumference as a function of r1 and r2. i am not sure how I should do this.

 

[synergix]

I have surface area= 2Ch(pi)r1r1. I need an equation for circumference and I don't think solving the previous equation for C is a good idea.

 

[Tedjn]

I have a few thoughts upon reading this question, some may be useful and some may not.

1. Your solution can rely on a special case of the isoperimetric problem.
   
2. For an intuitive way to see that the circle has smallest circumference, see marcusl's first comment.
   
3. One way to prove this directly may be to approximate the circumference of an ellipse with the perimeter appropriately symmetric, stretched regular polygons, and compare the perimeter to that of the regular polygon approximating the same area circle.

The last choice appears to me at the moment being tricky to apply. However, you are right that this is the key result. Finding the circumference of the ellipse directly is, as you've found, intractable.