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Digging the shortest trench to uncover a pipeline
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Imagine a straight run of pipeline which is buried
underground at a depth of 2 meters. Imagine also, that the pipeline
passes somewhere through a circular plot of land but that the exact
location of the pipeline is not known.
How would one go about
finding it? This problem is discussed in detail in
article by Ian Stewart
on page 206 of the September 1995 issue of Scientific
American .
One strategy would be to dig a ditch 2 meters deeps around
the perimeter of the entire circular plot of land. Since the pipeline
is known to pass under this area, the ditch is uncover part it. If the
circular plot ofland has a 1 kilometer radius then the proposed ditch
will be approximately 6.28 km. long. Is it possible to dig a
shorter ditch which is also certain to discover the location of the
pipeline?
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 Steprans

email address: steprans@mathstat.yorku.ca

Department of Mathematics and Statistics.

Ross 624 South, ext. 33952

York University

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