A beverage company must design a closed right circular can that holds exactly of juice
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A beverage company must design a closed right circular can that holds exactly 1 L of juice. The aluminium used for the cylindrical side wall costs $0.004 per square centimetre, while the material for each circular end costs $0.006 per square centimetre. Shipping adds $0.001 per centimetre of height per can. Provide complete analytic solutions for the following tasks.
(a) Express the total cost C in terms of the can's radius r and height h, in centimetres.
(b) Use the volume constraint to eliminate one variable, obtaining C as a function of a single independent variable.
(c) Compute drdC 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 yields the global minimum, including endpoint considerations.