A beverage company must design a closed right circular can that will hold exactly 1 liter of juice. The aluminum used for the cylindrical side wall costs \0.004persquarecentimeter,whilethematerialforeachofthetwocircularendscosts0.006 per square centimeter. In addition, shipping the cans adds \$0.001 per centimeter of height per can.
Provide complete analytic solutions for the following tasks.
(a) Express the total cost C (manufacturing plus shipping) in terms of the can’s radius r (cm) and height h (cm).
(b) Use the volume constraint to eliminate one variable, obtaining C as a function of a single independent variable.
(c) Compute drdC (or dhdC, depending on your choice in part (b)) and find all critical points in the feasible domain.
(d) Determine the radius and height that minimize the total cost, and calculate that minimum cost to the nearest tenth of a cent.
(e) Explain why the critical point found in part (c) yields the global minimum, explicitly addressing any endpoint considerations.