# Mathematical Methods - Cables.

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:

The first two equations, those for the “C” and “V” plans, are fairly easy to come by, through visual analysis of the diagrams. The “Y-plan” on the other hand, requires a much greater deal of user input and manipulation.

C - plan:  (when l is length of cable required and “a” is the distance from either town A or C to town B, as the crow flies.)

V - plan:

Y - plan:

Proof for Y - plan:

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