A beverage company wishes to design a right circular can that will hold exactly 1000 cm3 of liquid. The can consists of a closed top and bottom of uniform thickness and a cylindrical side wall made from the same material. The company would like to minimize the total amount of material used.
(a) Express the total surface area A(r) of the can as a function of the radius r of its base. Define all variables you introduce.
(b) Find all critical points of A for r>0. Show all differentiation steps.
(c) Determine whether the critical point you found corresponds to a minimum. Use an appropriate derivative test and provide a clear justification.
(d) Calculate the corresponding height h and the minimal surface area. Interpret the relationship between the height and the diameter of the can in one sentence.