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Calculus and Beyond Homework Help
Maximizing the volume of a cylinder
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[QUOTE="CrosisBH, post: 6331960, member: 643908"] [B]Homework Statement:[/B] 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. [B]Relevant Equations:[/B] [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! [/QUOTE]
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Maximizing the volume of a cylinder
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