Fall 2001


The Bailey Challenge

Two rectangles are shown in the figure.  The length of the large rectangle is twice its width.  The small rectangle has two of its sides along adjacent sides of the large rectangle and a vertex on the diagonal.  Express your answers to the following problems in terms of the width W of the large rectangle.


Problem 1. If the small rectangle is a square then find the ratio of the area of the square to the area of the large rectangle.

Problem 2. Find ratio of the areas of the two rectangles when the small rectangle has maximal area.

Send your solutions to Herb.Bailey@rose-hulman.edu or to Herb Bailey, Math. Dept., Rose-Hulman, 5500 Wabash Ave., Terre Haute IN 47803.

Solvers of the Papus problem were:

Alumni: B. Barrick, 1941; W. Soudriette, 1943; D. Mott, 1943; J. Hurt, 1948; A. Schairbaum, 1949; C. Cook, 1949; A. Junker, 1950; W. Rinker, 1951; C. Hirschfield, 1954; J. Moser, 1956; C. Cooper, 1956; H Brown, 1957; A. Mahler, 1971; M. Bailey, 1976; S. Warner, 1978; J. Slupesky, 1979; C. Abdnour, 1989; B. Burger, 1991; D. Devore, 1991; T. Anderson, 1992; D. Bailey, 1993; M. Pilcher, 1998; G. Lara, 2000; J. Briggs, 2001; C. Lehman, 2005; and E. Tollefson, 2005.

Friends: C. Drake

 The number of solvers was down a bit this time, perhaps spring fever.  Some missed the problem of balancing the trapezoid by assuming that the areas on either side of the balance line must be equal.  The correct requirement is that the moments must be equal.  A sketch is shown in the figure.  The moment of the rectangle around the vertical dashed line is (3)(x)(x/2).  The moment of the triangle about the dashed line is 1/2(3)(5-x)((5-x)/3).  Equating these two moments and solving the resulting quadratic gives x ~= 1.83.

 

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