It may at first seem that the answer is surely negative; in other words, digging a circular ditch around the perimeter is the most efficient means of discovering the location of the pipe line. After all, digging a ditch of smaller radius may fail to detect a ditch which passes through only a small part of the circular area. The same is true for a ditch dug along a diameter. Before looking at a solution, you should take some time to see if you can discover how to dig a shorter ditch which is certain to discover any straight pipeline passing through the circular area. (Hint: The length of this ditch is approximately 5.14 km.)
However, there may be even shorter solutions. One possible approach to finding an even shorter solution is to consider a trench composed of a circular part and three straight segments . First, convince yorself that any such trench will actually detect all straight pipelines. Then, find a particular configuration whose total length is less than 5.14 km. Is this the best possible? See Scientific American February 1996 page 125 for a strategy for constructing an even shorter trench which uses a circular segment connected to two tangent line segments as well as two disjoint line segments. Even this is not known to provide a minimal solution.
Before looking at the
solution, take some time to think how you would
go about solving this problem. This will help you understand the final
offered solution. You may even come up with a better solution.
Instructor
Juris Stepransemail address: steprans@mathstat.yorku.ca
Department of Mathematics and Statistics.
Ross 624 South, ext. 33952
York University
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