Optimization Problem.

[ffuh205]

I am really at a loss here. This is what I need to do. First, I need to explain a "strategy" for solving Optimization problems. Then I have to use that strategy to solve this really complication word problem. I have a company that wants to construct a wooden box with square base and open top which will contain exactly 16 cubic feet of volume. It costs P cents per square inch to build the sides and Q cents per square inch to build the bottom. What are the dimensions and Cost of the most economical box? (Minimum Cost)

 

[ffuh205]

I forgot to say that P = 2, & Q = 3.

 

[toidayma]

Well, let x = the base side, then the cost for bottom is P*x^2. Since the volume is 16, then the height of the side is h = 16/x^2, and the area of the four sides is 4*(16/x^2)*x = 64/x, therefore the cost of the sides is 64*Q/x
The function of the cost for one wooden box then is f(x) = P* x^2 + 64*Q/x with x > 0.
With x > 0, We have f'(x) = 2*P*x -64Q/x^2, f'(x) = 0 when x^3 = 32Q/P.
With this only value of f'(x) = 0, we can be sure that at this value, you can have the least cost, which is f(x) at x = (32Q/P)^(1/3).