Summer 2004

Bailey Challenge

By Professor Emeritus Herb Bailey

We all have money problems but few have equilateral triangle problems.  Prob. 1 is an example entrance exam question at Rose Poly in 1883.  I am hoping that somebody can solve it.  Prob. 2 is a classic money problem and I am hoping that many of you have not seen it.  Prob. 3 is from a recent SAT exam.  Prob. 4 is similar to 3 but a bit more difficult.

I have never taken a college entrance exam and do not plan to.  As a consolation to any of you with low SAT scores, it is not clear that they have anything to do with success in of out of college.  Some years ago, I made scatter plots of SAT scores versus GPA for Rose freshmen and found very little correlation.

Problem 1. Find the avails of a note for $500 discounted at a bank for 3 months at 8%.
Problem 2. Three people check into a hotel.  They pay $30 to the manager and go to their room.  The manager finds out that the room rate is $25 and gives $5 to the bellboy to return.  On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person.  Now each person paid $10 and got back $1.  So they paid $9 each, totaling $27.  The bellboy has $2, totaling $29.  Where is the remaining dollar?
Problem 3. Triangle ABC is equilateral and the sides of the inscribed triangle DEF are perpendicular to the corresponding sides of ABC as shown in Figure 1.  Find the ratio of the area of ABC to the area of DEF.
Problem 4. A rectangle is inscribed in an equilateral triangle ABC, with one side of the rectangle along AB as shown in Figure 2.  Find the ratio of triangle area to rectangle when the rectangle area is minimal.  Full credit for solving with or without calculus.  Extra credit for solving with and without calculus.

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 'spring problems' are listed.  Those who solved the bonus problem had many ingenious ways to communicate their solutions.  I made a paper cube as shown in the figure where the three voids are along a body diagonal and are colored white.  I tried to form the solution on my Rubik's cube without success.  You might give it a try.

Alumni: B. Barrick, 1943; H. Payne, 1947; J. Hurt, 1948; C. Cook, 1949; B. Powell, 1951; J. Lamberermont, 1951; D. Camp, 1955; C. Cooper, 1956; J. Chinn, 1956; A. Sutton, 1956; D. Bailey, 1959; J. Snyder, 1962; J. Lafuze, 1967; J. Born, 1970; W. Pelz, 1971; D. Willman, 1972; R. Smith, 1973; P. Chilson, 1974; M. Bailey, 1976; R. Priem, 1979; M. Clouser, 1979; J. Slupesky, 1979; B. Wade, 1983; M. Lancaster, 1987; D. Johnson, 1987; S. Johnson, 1988; C. Abdnour, 1989; J. Jachim, 1989; R. Pogliano, 1989; G. Tyrell, 1990; G. Heimann, 1990; B. Burger, 1991; J. Tindall, 1991; D. Hector, 1992; C. Tracy, 1997; M. Pilcher, 1998; B. Monacelli, 2000; B. Creel, 2000; P. Reskel, 2000; L. VanSchoiack, 2002; D. Harrington, 2002; K. Bopp, 2003

Friends: B. Talbot, R. Lindberg, C. Brown, J. Martin, S. Compton, L. Gainter, E. Mayhew, H. Novak, E. Mahler, M. Rosene