Optimization in a Crooked Trapezoid

Figure ABVU consists of three straight line segments and a circular arc. AU||BV, AU⊥AB. Lines tangent to the arc form trapezoids ABDC.

setup for the Optimization in a Crooked Trapezoid problem

Find the tangent that maximizes the area of ABDC.

Solution

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Copyright © 1996-2018 Alexander Bogomolny

Figure ABVU consists of three straight line segments and a circular arc. AU||BV, AU⊥AB. Lines tangent to the arc form trapezoids ABDC.

setup for the Optimization in a Crooked Trapezoid problem

Find the tangent that maximizes the area of ABDC.

The area of any trapezoid ABDC can be found from AB×(AC + BD)/2. In this formula an factor (AB) - the height of the trapezoid - is fixed. The problem then is equivalent to maximizing the half-sum of the bases.

solution for the Optimization in a Crooked Trapezoid problem

In a trapezoid the half-sum of the bases is equal to the midline - the line parallel to the bases and half way between them. Let M be the midpoint of AB, N on the arc such that MN⊥AB. The tangent at N solves the problem. Indeed, all other tangents cross MN below N so that the corresponding midline is shorter than MN.

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  • A Problem with a Magical Solution from Secrets in Inequalities
  • Leo Giugiuc's Optimization with Constraint
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  • A Cyclic Inequality With Constraint in Two Triples of Variables
  • Two Problems by Kunihiko Chikaya
  • An Inequality and Its Modifications
  • A 2-Variable Optimization From a China Competition
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    Copyright © 1996-2018 Alexander Bogomolny

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