- #1

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- Homework Statement
- Find the ratio of R (radius) to H(height) that will maximize the volume of a right circular cylinder for a fixed total surface area.

- Relevant Equations
- [tex] \frac{\partial f}{\partial q_k} - \frac{d}{dx} \frac{\partial f}{\partial q_k '} + \sum \lambda_k (x) \frac{\partial g_k}{\partial q_k} = 0[/tex]

[tex] S = 2\pi RH + 2\pi R^2 [/tex]

[tex] V = \pi R^2 H [/tex]

Note this is in our Lagrangian Mechanics section of Classical Mechanics, so I assume he wants us to use Calculus of Variations to solve it.

The surface area is fixed, so that'll be the constraint. Maximizing volume, we need a functional to represent Volume. This was tricky, but my best guess for it is

[tex] V = \int dV = \int_{0}^{H} \pi R^2 dH [/tex]

That way it can be represented by a single integral.

[tex] f = R^2 [/tex]

Plugging the stuff into the Euler-Lagrange equation and simplifying I get (I can show my work here if it's needed)

[tex] (2\lambda + 1)R + \lambda H = 0 [/tex]

I'm just stuck here. I don't know how to proceed. Calculus of Variations is still very new to me. Thank you!

The surface area is fixed, so that'll be the constraint. Maximizing volume, we need a functional to represent Volume. This was tricky, but my best guess for it is

[tex] V = \int dV = \int_{0}^{H} \pi R^2 dH [/tex]

That way it can be represented by a single integral.

[tex] f = R^2 [/tex]

Plugging the stuff into the Euler-Lagrange equation and simplifying I get (I can show my work here if it's needed)

[tex] (2\lambda + 1)R + \lambda H = 0 [/tex]

I'm just stuck here. I don't know how to proceed. Calculus of Variations is still very new to me. Thank you!