Chapter 14

G54. Team A beats F, and ties C; team B beats A, C and F; team C beats E and F; team D ties A and beats F; team E beats A and F; team F beats C and ties D. Rank these teams. If two teams tie, enter 0.5 in the communication matrix instead of 1.

G55. Suppose your group conducts an experiment at a local department store. You walk up a rising escalator and you take one step per second to reach the top in 20 seconds. Next, you walk up the same rising escalator at the rate of two steps per second and this time it takes 32 steps. How many steps would be required to reach the top on a stopped escalator?

G56. Two ranchers sold a herd of cattle and received as many dollars for each animal as there were cattle in the herd. With the money they bought a flock of sheep at $10 a head and then a lamb with the rest of the money (less than $10). Finally, they divided the animals between them, with one rancher obtaining an extra sheep and the other the lamb. The rancher who got the lamb was given his friend's new watch as compensation. What is the value of the watch?

G57. Problem G34 in Algebra Suppose your group has just been hired by a company called Alco. You are asked to analyze its operations and make some recommendations about how it can comply at a minimum cost with recent orders of the Environmental Protection Agency (EPA).

To prepare your report, you study the operation and obtain the following information:

• Alco Cement Company produces cement.
• The EPA has ordered Alco to reduce the amount of emissions released into the atmosphere during production.
• Alco wants to comply, but wants to do so at the least possible cost.
• Present production is 2.5 million barrels of cement, and 2 pounds of dust are emitted for every barrel of cement produced.
• The cement is produced in kilns that are presently equipped with mechanical collectors.
• To reduce the emissions to the required level, the mechanical collectors must be replaced either by four-field electrostatic precipitators, which would reduce emission to 0.5 pound of dust per barrel of cement, or by five-field precipitators, which would reduce emission to 0.2 pound per barrel.
• The capital and operating costs for the four-field precipitator are 14ó per barrel of cement produced; for the five-field precipitator, costs are 18ó per barrel.
• To comply with the EPA, Alco must reduce particulate emission by at least 4.2 million pounds. *

Use mathematical modeling to write your paper. Mathematical modeling involves creating equations and procedures to make predictions about the real world. Typical textbook problems focus on limited, specific skills, but in the real world you need to sift through the given information to decide what information you need and what information you do not need. You may need to do some research to gather data not provided.

G58. Knot Theory Get a piece of string with two free ends, and tie those ends together with a knot. Some knots that you can tie will hold the ends of the string string together and other knots will not (no pun intended!). In mathematics, there is a branch of mathematics known as knot theory. Mathematically, a knot is defined as a closed piecewise liner curve in R cubed. Two or more knots together is known as a link. Knots can be cataloged according to the number of crossings (ignoring mirror reflections). There is only one knot with crossing number three (called the cloverleaf knot), one knot with crossing number four, two with crossing number five, and three with crossing number of six.

* This research project is adapted from R. E. Kohn, "A Mathematical Programming Model for Air Pollution Control, " Science and Mathematics, June 1969, pp. 487-499.

1. How many knots are possible with crossing number of seven?
2. How many knots are possible with crossing number of eight?

Write a paper on knot theory.