You have a can with the shape of a cylinder. The bottom and top part of the can both has a thickness twice the thickness of the walls of the can. In order to minimize the cost of material what should the relation between the diameter and the can height be if the can has a constant volume?
I guess the way to solve this problem is to write an equation f(x)= ... and then differentiate it and set that equal to zero f'(x) = 0 to find the lowest point on the curve. (Sorry for my poor math language)
Can someone tell me how to solve this problem please?
Sounds like you have an inkling of what you need to do.
Surface area of a cylinder $2\pi r^2+ 2\pi r h$
But that is an ordinary cylinder, You need one with double thickness top and bottom.
Objective: minimize $f(r,h) = 4\pi r^2 + 2\pi r h$
Constrained by: $V = \pi r^2h$ ($V$ is a constant)
Use the constraint to find $h$ as a function of $r$ (or $r$ as a function of $h$), and substitute into to the objective function. This will make the objective function of $r.$
Differentiate the objective, set the derivative equal to $0$ and solve for $r$ (or $h$).
Then return to the constraint, or the function you derived from the constraint to find $h.$