The rush-hour traffic flow | Mathematics homework help

The rush-hour traffic flow for a network of for one-way streets in a city is shown in the figure. The numbers next to each street indicate the number of vehicles per hour that enter and leave the network on that street. The variable x1, x2, x3 and x4 represent the flow of traffic between the four intersections in the network.

 

(a)   For a smooth traffic flow, the number of vehicles entering each intersection should always equal the number leaving. For example, since 1000 vehicles enter the intersection of 1st Street and 3rd Street each hour and x1 + x3 vehicles leave this intersection, we see that x1 + x3 = 1000. Find the equations determined by the traffic flow at each of the other three intersections.

(b)   Find the solution to the system in (a).

(c)   What is the maximum number of vehicles that can travel from 3rd Street to 4th Street on 1st Street? What is the minimum number?

(d)   If traffic lights are adjusted so that 700 vehicles per hour travel from 3rd Street to 4th Street on 1st Street, determine the flow around the rest of the network.

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