A shipping company must build an open-top rectangular box with a square base to carry small parcels. The interior volume of the box must be 32000cm3. All four side panels and the base are to be constructed from corrugated cardboard that costs $0.012 per cm2. No lid is required.
(a) Let x (cm) represent the length of one side of the square base and h (cm) the interior height of the box. Derive an expression for the total material cost C(x,h) of one box.
(b) Write the volume requirement as an equation in x and h and solve for h in terms of x.
(c) Substitute the expression from part (b) into C(x,h) to express cost as a function of x only. Determine the value of x that minimizes the cost. Show complete calculus justification.
(d) Confirm that the dimension found in part (c) yields a minimum. Then compute the corresponding height h and the minimum cost, reporting each to the nearest tenth of a centimeter and the nearest cent, respectively.
(e) Suppose the company decides to add a lid made of the same material, increasing the cost. Without performing any new calculus, explain qualitatively whether the optimal value of x will increase, decrease, or remain the same. Justify your answer.