A cylindrical container costs $3.00 per square foot for the sides and $4.00 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.

To solve this problem, you must find the minimum value for one of the variables in a specialized from of the formula the surface area of a cylinder, which is known to be

A = 2 * pi * r ^2 + 2 * pi * r * h

The specialized form for the price, based on the values in the given problem is:

P = (2 * pi * r^2) * 4 + (2 * pi * r * h) * 3

r and h are related in the formula for the volume, which is:

V = h * pi * r^2

Since you know that V = 100, you can solve for either of the variables in the volume function:

100 = h * pi * r^2
h = 100 / (pi * r^2)

Substitute that into the formula with values above, and you get

P = (2 * pi * r^2) * 4 + (2 * pi * r * (100 / (pi * r^2))) * 3

Now you just have to find the value of r that gives the smallest value for A