A manufacturer wishes to build an open-top rectangular box with a square base that will hold exactly 32 000 cm3 of liquid disinfectant. The bottom of the box is to be made of a special corrosion-resistant material costing $4 per cm2, while the four sides are made of a different material costing $2 per cm2. Let x cm be the length of a side of the square base and let h cm be the height of the box.
(a) Write an expression for the total construction cost C of one box in terms of x and h. Include the contribution from each portion of the box.
(b) Use the volume requirement to rewrite C as a function of x alone.
(c) Determine the value of x that minimizes the total cost, and then compute the corresponding height h.
(d) Justify that the dimensions found in part (c) indeed yield a minimum total cost, and state the minimum possible cost to the nearest dollar.