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 x_1, x_2, and x_3 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 x_1 + x_2 cars leaving A. Therefore, x_1 + x_2 = 500. For intersection B, we have 50 + x_2 cars per hour entering the intersection and 100 + x_3 cars leaving the intersection. Thus 50 +x_2 = 100 +x_3, or x_2 - x_3 = 50. Applying the same reasoning to C. we have x_1 +x_2 = 450. These equations result in the system of equations: {x_1 + x_2 = 500 (1) x_2 - x_3 = 50 (2) x_1 + x_3 = 450 (3) 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 6[e] to 8[a] cars per hour, what is the estimated traffic flow between C and A, and between B and C?

Explanation / Answer

At junction A:
Incoming vehicles = outgoing vehicles
x3 + 200 = x1 + 165
x1-x3 = 35 - equation 1

At junction B:
Incoming vehicles = outgoing vehicles
x1 + 50 = x2 + 100
x1-x2 = 50 - Equation 2


At junction C:
x2+200 = x3+185
x3-x2 = 15 -equation 3

Assuming that 6[e] meant 60 and 8[a] meant 80;
x1 lies between 60 and 80 (60 and 80 included) cars per hour

x2 = x1 - 50
so x2 will be in the range of [10,30] cars per hour
So, the flow of cars from point B to C will be from 10 to 30 cars per hour

x3= x1-35
so x3 will be in the range of [25,45] cars per hour

So, the flow of cars from point C to A will be from 25 to 45 cars per hour

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