minimizing weight of a tank using optimization in calculus?

Question

You have a 500 ft^3, square-based, open-top, rectangular steel holding tank, Find the dimensions for the base and height that will make the tank weigh as little as possible.
So I know I have to use derivitives and the volume and area formulas, but they're not working. I know the answer is 10 x 10 x 5, but I can't get it. The formulas I have are: V(500)=lwh or s^2h and A=s^2+4sh. I can only assume my equations are wrong, but what's wrong with them??? Any help would be very much appreciated

Answer 1

Assuming the weight of the tank is based on the weight of the walls, and the walls of the tank are of equal thickness; then the weight of the walls is proportional to their total area. You minimize the weight by minimizing the area.

Since s^2*h = 500, h is a function of s, specifically h = 500/s^2.

Therefore A = s^2 + 4s (500/s^2) = s^2 + 2000/s

Now you can plot A as a function of s. I don't know if that will help or not, last time I did this was 20 years ago.

Of course, if you *really* wanted to optimize the tank you'd make it a cylinder or sphere.