Abstract:

Given a regular polygonal paper inscribed in a unit circle, the paper is cut along its radii and each division (consisting of one or more subdivisions) is made into a cone. These cones are allowed to be slanted to obtain a greater capacity. The purpose of this study is to maximize the total capacity of cones made from the paper over all ways of divisions.

The methodology in this report is streamed into two parts – minimax strategy and bounds by inequalities. For triangular paper, the rims of cones are parameterized before their water depths are expressed explicitly. The capacities of cones are maximized over angles of slant. Different ways of division are compared to and out the optimal solution. Probing into general cases, various inequalities are set up analytically and exhaustively to bound the total capacities for comparisons.

To obtain the greatest capacities, cones made from one sub-division should be slanted but those from multiple sub-divisions should be held vertically. For a polygonal paper of six or more sides, it should be divided into two divisions, each comprising two or more sub-divisions with a central angle ratio of 0.648:1.352, approaching the way of division in circular paper.

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