Stuart Ball Betty Taylor Brenda Weddell
Materials: Graph paper, ruler, protractor, compass (possibly construction paper and scissors)
Goals: To apply the relationships between circles and triangles, and the right triangle relationships (especially the Pythagorean Theorem) to "real-world" situations.
Time Required: 30 - 60 minutes
Background: Students should be familiar with the Pythagorean Theorem, and be able to relate the parts of a triangle to a circle circumscribed about it; if students use trial-and-error to solve the problems, no real geometry background is necessary.
Technology Required: Scientific calculator
Central State Telephone Company uses a standard size cable 16 mm in diameter for all of its underground and underwater work. Before being buried or sunk, these cables are usually inserted into a plastic pipe for protection. Of course, if a single cable is inserted into a pipe, the diameter of the pipe must be at least 16 mm, the same as the cable. And if two cables are inserted into a pipe, the diameter of the pipe must be at least 32 mm, since the cables will lie side-by-side no matter what. The interesting problems occur when Central State wants to insert 3 or 4 cables into a single pipe, as shown in these two diagrams:
What is the minimum diameter pipe that will allow 3 cables to be inserted? What is the minimum diameter pipe that will permit 4 cables to be inserted? (The diagrams may give you some hints on how to proceed.)
Your solution should include:
1. your answer(s), with appropriately labeled diagrams where necessary,
2. a list of assumptions and limitations that are part of your solution, and
3. a step-by-step description of your solution, giving reasons for each step.
Be prepared to present your solutions to the entire class!
Students may use two different techniques to solve this problem. First, if you are using it as an introduction to a unit on triangle relationships, they may use trial-and-error to find the answers. You might want to provide them with construction paper and scissors and have them try a variety of circles until they find the size that fits around the cables in each case.
Of course, you can also use this exercise after you have presented the material. In this case, the students should be able to use more analytic methods to determine the appropriate sizes of the pipe. The equilateral triangle shown in the 3-cable pipe, and the square shown in the 4-cable situation, should give students a place to start. (If you don't want them to have this much of a hint, you may delete the triangle and square. You may even want to remove the diagrams completely!)
The minimum diameter of the pipe for the 3-cable arrangement is 643/3 + 16 mm, or approximately 53.0 mm.
For the 4-cable arrangement, the minimum diameter of the pipe is 162 + 32 mm, or approximately 54.6 mm.
By the way, there is no known formula for determining the minimum diameter for the general case of n cables. However, a great deal of research has been done into such "dense packing" problems.
Funded in part by the National Science Foundation and Indiana University 1995