The Mug That Holds the Most Liquid

An important goal in engineering is to reach maximum function with minimum resources. This Demonstration answers the question: how do you design a cylindrical mug to contain the maximum liquid for a given volume of material?



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Snapshot 1: the objective function of liquid volume, which is the volume of a cylinder , where is the radius and is height
Snapshot 2: the constrained function of the mug volume, which is the sum of the outer shell and the bottom
Snapshot 3: the optimal solution, which always occurs when the mug's inner radius equals the inner depth (); the mug holds the most liquid possible with the given amount of material
Snapshot 4: the current and optimal design shown in a 2D contour plot where the liquid volume is the target function and the pink line is the constrained function, the mug material; the yellow point indicates the liquid volume for the current mug parameters and the red point is the optimal design
(For a closed container like an oil tank, the constrained function is slightly different. In that case the optimum occurs when .)
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