Minimize Cost Example

Business Calculus · Axiom Academy

Classic optimization: minimizing material for a can A beverage company wants to design a cylindrical can that holds exactly 355 mL (standard soda can volume) while using the minimum amount of aluminum. Find: The dimensions (radius and height) that minimize the surface area. Let r = radius and h = height of the cylinder What we want to minimize: Surface Area (amount of material) Constraint: Volume must equal 355 cm³ Substitute into the surface area formula: Surface area as a function of radius only Take the derivative with respect to r: Notice that the optimal height is exactly twice the optimal radius: The optimal can has h = 2r (height = diameter) When h = 2r , the can is exactly as tall as it is wide (diameter). This creates the most "spherical-like" cylinder possible, and spheres minimize surface area for a given volume. Actual soda cans are taller and thinner than this optimal design. Why? Ergonomics: easier to hold a taller can Shipping: taller cans pack more efficiently in boxes Marketing: taller cans appear larger to consumers Real optimization often involves multiple objectives! You've mastered constrained cost minimization! Based on the example above, which approach is correct?

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