 #1
CrosisBH
 27
 4
 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 EulerLagrange 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 EulerLagrange 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!