Mathematical Methods Portfolio #2 2003
Type 3 (Modelling): Cables
Everyday, people are faced with problems that can be solved through the use of mathematics, particularly in the fields of design and construction, where geometry, trigonometry and algebra become very important. This is the point at which this portfolio piece becomes very relevant. The task at hand is as follows:
Engineers put forward three plans to provide a telephone link between three towns: A, B and C. An isosceles triangles is formed by these towns with the distance AB = BC = a km and the distance from A to C = 2 km.
Engineers wish to use the least amount of cable to join the three towns.
C - Plan V- Plan Y - Plan
The objective of the portfolio is to ascertain, for varying positions of town B, and the resultant differing lengths of a, which plan will in the end provide the link using the least amount of cable.
NB. A must be greater than 1 in all circumstances, as all towns must be linked and the plans are all based on triangles.
To begin with, one must find an expression for the cable length for each of the plans in terms of the distance a: