A cylindrical container costs $2.00 per square foot for the sides and $3.00 a square foot for the top and bottom. The container must hold 100 cubic feet of material. What are the dimensions of the most economical container.

The surface area of a cylinder(with top and bottom) is given by:

S=\underbrace{2{\pi}r^{2}}_{\text{top and\\bottom}}+\overbrace{2{\pi}rh}^{\text{cylinder }}... [1]

Volume is: V={\pi}r^{2}h=100... [2]

Remember your costs:

Using [1], the cost of the material would then be:

6{\pi}r^{2}+4{\pi}rh... [3]


Now, solve [2] for h and sub into [3]. You will then have the function in terms of r alone.

Differentiate, set to 0 and solve for r. Once you have r, h will follow.