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A rectangular container is open at the top and must have a volume of 10 m3. The material for the sides costs C dollars per m2, while the material for the bottom costs 2C dollars per m2. Find the optimal dimensions using Lagrange multipliers so as to minimize total cost of the container and what are the units for λ?
I am having trouble approaching the problem, only knowing that we start off with V= Lwh = 10 for the volume of the box. While cost = 2(wl) + 1(2wh + 2lh).
Your constraint is on the volume, as you said $$ g(l,w,h)=lwh=10 $$ And you are seeking to minimize the function $f(l,w,h)=2c(wl)+c(2wh+2lh)$.
So your set up is then $$ \nabla f=\lambda\nabla G\\ \Rightarrow 2cw+2ch=\lambda wh\\ 2cl+2ch=\lambda lh\\ 2cw+2cl=\lambda wl $$ Where your answer will be in terms of $c$.
