A beverage manufacturer is designing a closed right circular aluminum can to hold 1L of soda, assuming 1000cm3 of liquid. The aluminum used for the sidewall costs 0.010 dollars per cm2, while the reinforced aluminum used for each lid (top and bottom) costs 0.025 dollars per cm2. Ignore any loss of material at seams.
(a) Let r be the interior radius (in cm) and h the interior height (in cm). Derive an expression C(r,h) for the total manufacturing cost, in dollars, of one can.
(b) Write the volume constraint in terms of r and h, then solve this equation for h in terms of r.
(c) Substitute the expression for h from part (b) into C(r,h) to obtain C as a function of r only. Determine the critical value of r that minimizes C(r). Show all calculus work, including derivative computations.
(d) Verify that the critical value found in part (c) yields a minimum total cost, and compute the corresponding height h.
(e) Interpret your result in context by stating the optimal radius, height, and minimum material cost per can to the nearest 0.0001 dollar.