Using Mathematical Optimization to Determine the Ideal Container Shape for Maximizing Volume and Minimizing Surface Area

This investigation explores how mathematical optimization can identify the ideal container shape to maximize volume while minimizing surface area for sea glass collection. The study aims to find the most efficient geometry for transporting specific quantities of irregular objects.

Using a fixed volume constraint, the methodology applies calculus to optimize three geometric models: a cylinder, a cone, and a rectangular prism. The approach includes first and second derivative tests, the "sin test" to verify local minima, and graphical validation using Desmos.

Results demonstrate that the cylinder is the most efficient shape, requiring the least surface area to contain the target volume. The paper is characterized by thorough critical point verification and a nuanced evaluation of real-world modeling limitations.