I'm doing questions from this page: http://tartarus.org/gareth/maths/tripos/IB/Variational_Principles.pdf and I'm doing Question 2013 1/I/6A
The question asks to find the cylindrical cup of least mass, where the base has density $k\rho$ and the curved side has density $\rho$. I've called the radius of the base $r$, and the height of the cup $h$.
The goal is to find $r$ and $h$ in terms of $V, k, \rho, \pi$
I've used Lagrange multipliers to get the three equations:
$(1) \space 3k\rho r^2 + 2h - \lambda h = 0$
$(2) \space 2\rho r - \lambda r^2 = 0$
$(3) \space V = \pi r^2 h$
I get the answers $r = 2\rho /\lambda$ and $\lambda = \frac{2\pi^2 h \rho^2}{V}$ But then when I put those values into $(1)$, I get a quartic that I don't know how to solve. This question is supposed to be short, so it shouldn't involve doing any complicated solving of quartic equations. Maybe there's a factorisation, but I can't see it.
Can somebody help?
Thanks