Traffic engineers use systems of equations to study the flow of traffic. The ana
ID: 2911509 • Letter: T
Question
Traffic engineers use systems of equations to study the flow of traffic. The analysis of traffic flow is based on the principle that the number of cars that enter and leave an intersection must be equal. Suppose the traffic flow for some one-way streets can be modeled by the diagram below, where the numbers and the variables represent the numbers of cars entering or leaving an intersection per hour.
Using this example, estimate the traffic flow for a roundabout presented below. The graphical model shows the number of cars per hour that are entering or leaving a roundabout. If the portion of the roundabout between A and B has an estimated traffic flow of from 69 to 82 cars per hour, what is the estimated traffic flow between C and A, and between B and C?
100 50 200 300 200 250Explanation / Answer
Answer :
The only usable inference from the example given in this question is that
Principle 1 - The numbers of cars going INTO an INTERSECTION = Number of cars going OUT of an INTERSECTION
An intersection is basically a meeting point of any two or more roads.
As can be seen in the coloured diagram, A, B, C are the points of intersection since the straight roads are meeting the cirular road at these points.
We hereby follow the same Principle 1
At intersection A we have :
Cars coming in from circular road = X3
Cars coming in from straight road= 200
Total cars coming into A= X3 + 200
Cars going out on Straight road = 165
Cars going away on circular road= X1
Total cars going out of A = X1 + 165
Using Principle 1
X3 + 200 = X1 + 165
X1 - X3 = 35 -----------eqn 1
At intersection B we have :
Cars coming in from circular road = X1
Cars coming in from straight road= 50
Total cars coming into B= X1 + 50
Cars going out on Straight road = 100
Cars going away on circular road= X2
Total cars going out of B = X2 + 100
X1+ 50 = X2 +100
X1 - X2 = 50 --- eqn 2
Only this much is required for our calculations. We dont require information for intersection C hence we will not waste time writing them.
Using Eqn 1 and 2 and putting limiting values of X1 as 69 to 82.
For X1= 69
X1 - X3 = 35
69- 35 = X3
34 = X3
and
X1 - X2 = 50
69-50 = X2
X2= 19
For X1 = 82
X1 - X3 = 35
82- 35 = X3
47 = X3
and
X1 - X2 = 50
82-50 = X2
X2= 32
Therefore X2 cars lie between 19 and 32
X3 cars lie between 34 and 47.
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