• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Mathematical Methods - Cables.

Extracts from this document...


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.

...read more.


Now that equations have been obtained for each of the models/plans, the use of technology, to create visual representations of the equations can now be used. Through this a deeper, clearer understanding can now be gained. One can quite obviously observe the differences between models/plans for the amount of cable required for the varying values of a.

In this graph, the y axis represents the total amount of cable used in the selected model/plan and the xaxis represents the varying values of a.

Thus, it can be deduced that the equation that possesses the lowest y value for any value of x will be the most effective model/plan for that specific x value.


Diagram 1.

Through a quick observation of the above diagram, it can quite easily stated, according to the previous deductions, that the most effective values of x (those which have the lowest values of y (use the least amount of cable)),for each of the plans, are:


...read more.



Proof using simultaneous equations:


Both of these equations cannot be solved algebraically with the techniques we have been taught at this stage of our education. Although, they can be solved through the implementation of graphics calculators where it can be found that in the modified Y-plan vs. the V-plan, a will equal image29.png and that in the modified Y-plan vs. Y-plan, a will equal 2.

The modified Y-plan does not ever seem to intersect with the C-plan, the modified y-plan always being a few units lower. This may not be the case at high values of a.

The modified Y-Plan provides the most effective use of cable for all possible values of a.

To conclude, the modified Y-plan provides the engineers with a telephone link which uses the least amount of cable possible. The diagrams used were drawn in Graphmatica and a majority of testing was done on a Texas Instruments TI-83 graphics calculator.

Mathematical Methods              Portfolio Piece 2-Cables (Type 3 – Modelling)                

...read more.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Core & Pure Mathematics essays

  1. 2D and 3D Sequences Project Plan of Investigation

    2b + c = 4 12a + 2b = 12 We can simplify this equation to: 6a + b = 6 My next calculation is below: N =3 _27a + 9b + 3c = 26 12a + 6b + 3c = 12 15a + 3b =14 (15a + 3b = 14)

  2. Best shape for gutter and further alegbra - using Excel to solve some mathematical ...

    Below is a snapshot of the excel spreadsheet showing optimum results. ?� w/2 w/2 Area 88 5 5 12.49239 89 5 5 12.4981 90 5 5 12.5 91 5 5 12.4981 92 5 5 12.49239 The formulae used to calculate the area of this cross section is given in appendix 2.

  1. MEI numerical Methods

    This will be significant later. However from this we can conclude that the false position method does not have a first order convergence. Method of bisection: If we apply the first order of convergence principles to this method we get the following: Unlike the secant and false position method, the method of bisection has a constant value of K, 0.5.

  2. Solutions of equations

    ROOT B ANSWER = 0.66667 (5 d.p.) ROOT A As mentioned earlier the method failed for Root A as there was no change of sign between f(0.4) and f(0.5). I know there is a root because of the graph and f(x) = 0 for x = (4/9) or 0.44444 (5.d.p).

  1. Although everyone who gambles at all probably tries to make a quick mental marginal ...

    Sum to infinity = when a is the first term in the sequence, and r is the ratio between terms. Thus = one has a 1 in 4 chance of making money when paying a $5 entry price. When compared to Case 1 and its odds of making money this game seems quite good.

  2. Math Portfolio Type II - Applications of Sinusoidal Functions

    6. State the range, domain, and period of the function representing the time of sunrise for Miami that you found in question 5. The domain of the function would be {x ? R | 1 ? x ? 365} and the range of the function would be {y ?

  1. Find methods of solving equations, which can't be solved algebraically.

    this root are [-0.5321, -0.5320] as such the solution is -0.55325 and the maximum error is +0.00005 or -0.00005. [2,3] 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 -3 -2.969 -2.872 -2.703 -2.456 -2.125 -1.704 -1.187 -0.568 2.9 3 0.159 1 2.8 2.81 2.82 2.83 2.84 2.85 2.86 2.87

  2. Sars Math Portfolio ...

    1, 2, 4, 8, 16 etc.) therefore even though our death figure for the end of week 4 is a logical figure, it does not take into account the nature factor. 4. There are different types of functions that can be used to model the first four weeks of data.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work