What is the optimal ratio of R/H to minimize the energy for a fixed volume?

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The discussion revolves around finding the optimal ratio of radius (R) to height (H) that minimizes the energy of a particle confined in a right circular cylinder with a fixed volume. The ground state energy is expressed in terms of R and H, with specific constants C1 and C2 derived from quantum mechanics principles. The participant expresses confusion about how to approach the problem and seeks guidance on relating the energy formula to the fixed volume constraint. They consider rewriting the dimensions in terms of new variables and reference a similar problem regarding maximizing volume, indicating a potential strategy to adapt. Ultimately, they arrive at a preliminary ratio of 1.85R = H, though they acknowledge the need to ensure the volume remains constant.
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Homework Statement


I can not find anything online or in my textbooks to help me with extremizing. please help...

I need to find a ratio of R/H that will minimize the energy for a fixed volume
I am given the ground state energy of a particle that is inside a right circular cylinder. and has a height H and radius R.

Homework Equations


ground state energy http://www.wolframalpha.com/input/?i=E = C_1/R^2 + C_2/H^2
C_1 is http://www.wolframalpha.com/input/?i=C_1 = h^2 / (2m) * (2.4048)^2
C_2 is http://www.wolframalpha.com/input/?i=C_2 = h^2 / (2m) * pi^2

The Attempt at a Solution



I don't understand the problem or how to begin. I have never done anything with energy of particles in cylinders. I don't know how ground state energy is affected by the height and radius of the container. I am sure everything I need is right in front of me, but I don't understand. I am not looking for someone to just give me the answer, but to walk me through it please. Please help!

I feel like I should start by rewriting h and r in terms of x and y
h = ds = sqrt (1 + x'^2)dy
and r...
 
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I think I have something. I found a similar problem that find the ratio R to H that maximizes volume for a right circular cylinder... maybe I can do that and just swap the formula E for the volume formula?

but would I plug in C1 and C2 at the beginning, or can I just wait till the end? what would i do with that?
 
I attempted to do a switcheroo and I got 1.2R = H
 
whoops I forgot that volume is fixed.
brb

ok I got 1.85R = H
 
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