[amriaf]

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.

[Nosnosna]

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