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Problem of the Week 1043

Queen Dido's Ghost

Find a function f (continuously differentiable) such that:

  • The domain of f is [0, 1]
  • The values of f(x) are nonnegative
  • f(0) = f(1) = 0
  • The integral of f from 0 to 1 is 1
  • The arc length of the graph of f is as small as you can make it
  • f is continuous and differentiable except at finitely many points.

Extra Credit: Do this with the value of the integral set to A instead of 1.

Note: While a proof of optimality is not called for, comments are welcome. It might be that there is no function that achieves a minimum length, just a shape that one can approach arbitrarily closely to. Of course, it is allowed to use piecewise functions. Those of you using Mathematica should know that there is a new Piecewise[ ] construction that is pretty elegant.

Source: The calculus text by Stewart has as a project the investigation of this problem with area 1. Of course, the problem title is a reference to Queen Dido, who wanted to lay out a string of length L against the (straight) coastline of the Mediterranean Sea so as to capture the largest area. The correct answer in that case is to place the string in the shape of a semicircle.

© Copyright 2005 Stan Wagon. Reproduced with permission.

8 November 2005