Traffic Flow Traffic engineers use systems of equations to study the flow of tra

Question

Traffic Flow

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.

Let x1, x2, and x3 represent the number of cars per hour that are traveling on AC, AB, and BC, respectively. Consider intersection A. There are 300 + 200 = 500 cars per hour entering A and x1 + x2 cars leaving A. Therefore, x1 + x2 = 500. For intersection B, we have 50 + x2 cars per hour entering the intersection and 100 + x3 cars leaving the intersection. Thus 50 + x2 = 100 + x3, or x2 - x3 = 50. Applying the same reasoning to C, we have x1 + x3 = 450. These equations result in the system of equations:

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 65 to 82 cars per hour, what is the estimated traffic flow between C and A, and between B and C?

Explanation / Answer

Consider junction A:

X3 + 200 = X1 + 165

X1 - X3 = 35

Consider junction B:

X2 + 100 = X1 + 50

X1 - X2 = 50

Consider junction C:

X2 + 200 = X3 + 185

X3 - X2 = 15

Suppose X1 = 65

solving above equations -->

X2 = 15

X3 = 30

Suppose X1 = 82

solving above equations -->

X2 = 32

X3 = 47

So estimated traffic flow between C and A which is X3 from 47 to 30 cars per hour

And estimated traffic flow between C and B which is X2 from 32 to 15 cars per hour

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