Arectangular box is to have a square base and a volume of 40 ft3. if the material for the base costs $0.37 per square foot, the material for the sides costs $0.05 per square foot, and the material for the top costs $0.13 per square foot, determine the dimensions of the box that can be constructed at minimum cost.

length
heigth
width

So hmm if you check the picture below, the area of the base is x*x and the top's is x*x and the sides, are 4 rectangles, each with area of h*x

thus 



now, the cost function, or C(x), well, we know how much area is going to be used in x-term, so let's just apply the cost to each



so.... now just get the derivative of C(x) and zero it out out and check the critcal points for any minima

Top right since it passes vertical line test