Chapter 4.3, Problem 97E

Traffic Wow. The rush-hour traffic flow for a network of four one-way streets m 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 variables $x_1,x_2\text{ and }{x}_3$ represent the flow of traffic between the four intersections in the network.

(A) For a smooth traffic the number of vehicles entering each intersection should always equal the number leaving. For example, since $1,500$ vehicles enter the intersection of 5th Street and Washington Avenue each hour and $x_1+x_4$ vehicles leave this intersection, we see that $x_1+x_4=1,500$. Find the equations determined by the traffic flow at each of the other three intersections.

(B) Find (the solution to the system in part (A).

(C) What is the maximum number of vehicles that can travel from Washington Avenue to Lincoln Avenue on $5^{th}$ Street? What is the minimum number?

(D) If traffic lights are adjusted so that $1,000$ vehicles per hour travel from Washington Avenue to Lincoln Avenue on 5th Street, determine the flow around the rest of the network.