So this is the exact problem:
What I understand from the problem is that the surface area S(r,h)=2πr² + 2πrh (according to what the surface area of a cilinder should be) and that the volume given by 2πr²h=1000 will be the restriction. When I apply lagrange i get the following system of equations:An industrial container is in the shape of a cylinder with hemispherical ends. The container must hold 1000 liters of fluid. Determine the radius r and length h that minimize the ammount of material used in the construction of the tank.
Solving that for λ I get λ=1\2r, and when I substitute that in the other equation, I get that r=0 which is ridiculous, because that would mean that in order to minimize the amount of material used for the tank the surface area would have to be zero! Obviously there's something wrong with the way I'm tackling the problem, so is there any way any of you guys could help?