Optimal Design Examples

Mathematical Modeling · Axiom Academy

EXAMPLE Optimal Design Examples Worked examples applying optimization techniques to real-world design problems A manufacturer wants to design a cylindrical can that holds exactly V = 500 mL of liquid. The can must have a circular top and bottom. What dimensions (radius r and height h ) minimize the amount of material used (surface area)? The surface area consists of two circular ends plus the curved side: The volume constraint fixes the relationship between r and h: Solve the constraint for h and substitute into the objective: Take the derivative and set it equal to zero: The second derivative test confirms this is a minimum. The optimal height is: Key Insight: The optimal can has height equal to diameter (h = 2r). This ratio minimizes material use for any fixed volume. For V = 500 mL, this gives r = 4.30 cm and h = 8.60 cm. Real soda cans are taller and thinner due to manufacturing costs, grip ergonomics, and shelf stacking considerations. If the manufacturer decides to use a thicker (and more expensive) material for the curved side than for the top and bottom, how would the optimal shape change? A farmer has 400 meters of fencing and wants to enclose a rectangular field that borders a straight river. No fence is needed along the river. What dimensions maximize the enclosed area? The fence covers two widths plus one length (river side needs no fence): Solve the constraint for y and substitute into the area formula: Take the derivative and find where it equals zero:

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