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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using simultaneous equations, get x by itself
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 x axis 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:
NB. At the intercepts of the graph, where two equations have the same values for x and y, both of the models/plans are the most effective, if the intercept has the lowest y value of the graph.
After much deliberation it was concluded that the best method by which to support the conclusions made above was the use of simultaneous equations. Through the use of simultaneous equations the value of x at the intersection at (1.51,3.02) can be proved.
Proof:
In this proof, V-plan will be solved simultaneously with the Y-plan.
Proof that V-plan is most effective for values of a less than 1.51. “a” will take a value of 1.
a = 1
Y – plan cannot take a value of 1 as it asymptotes at x = 1, therefore never actually touching x = 1. “a” can never = 1in the equation for the Y – plan.
V-plan has the lowest possible y value (amount of cable) when x is equal to 1 therefore it is the most effective plan to use when a is less than 1.51.
Proof to support the conclusion that Y-plan is the most effective plan to use for values of a between 1.51 and 3.55. Simultaneous equations will be used again.
In this proof C-plan will be solved simultaneously with Y-plan.
At this stage of our proof the equation becomes highly complex and requires the use of a graphics calculator to solve it. Further proof can be provided by substituting the value of 2 for a.
Y-plan is the most effective plan to implement for these values as it possesses the lowest value (amount of cable used.)
Finally, proof that C-plan is the most effective plan to implement if the distance between either town A or C and town B is greater than 3.55. The value 4 will be substituted for a, to prove this.
C-plan provides the lowest possible y value when a is substituted for 4 and thus, it is the most effective.
A modification to the Y – plan was proposed, where each of the angles would be locked at 120º. As a result of this, the distance between the towns becomes unequal, thus the an x value must be changed to y.
Using this new information a new equation for Y-plan can be attained.
Modified Y-plan = Length 2 = 2x + y
Below: The new equation graphed against the original plans.
Diagram 2
After a quick observation of the new graph, one can quite obviously see that the new proposed plan () is clearly as, or more effective as the other plans.
There are two occasions in which two or more plans possess the lowest y value for the same x value. This is called a tangent. The tangents are formed between V-plan and the revised Y-plan and between both of the Y-plans. Through the use of a graphics calculator these tangents can be found. They are:
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 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.
