5. The principle of equal flow rates into and out of a junction applies to many

Question

5. The principle of equal flow rates into and out of a junction applies to many applications of networks, including communications systems, water systems, and even the circulatory system in the human body Traffic Control: If traffic is to keep moving, during any period of time the number of cars entering an intersection must equal the number of cars leaving that intersection. The figure below shows the intersection of three one-way streets. The numbers in the figure denote the number of cars per minute that travel in the direction shown. If the traffic is to keep moving, at each intersection the number of cars entering per minute must equal the number of cars leaving per minute. The following system of linear equations involving x, y, and z models this situation. 10 cars/min 6 cars/min Sunset Drive x+10-14+y 2+12 = 6 + x y+6=8+2 12 cars/mi 14 cars/min Canal Street Orange Ave 6 cars/min 8 cars/min a. Solve the system of equations using Gauss-Jordan Elimination b. Write the solution in terms of z c. If construction limits z (Orange Ave) to no more than 2 cars per minute, how many cars per minute must pass between the other intersections to keep traffic flowing? Note the restrictions on z. The range of z will be from the minimum to the maximum number of cars allowed.] Orange Ave Canal Street Sunset Drive 0

Explanation / Answer

x+ 10 = 14 + y

z + 12 = 6 + x

y + 6 = 8 + z

x - y + 0*z = 4 ---------------------- (1)

-x + 0*y + z = -6 -----------------------(2)

0*x + y -z = 2 -----------------------------(3)

Augmented matrix is :

1 -1 0 | 4

-1 0 1 | -6

0 1 -1 | -2   

Step 1) R2 = R2 + R1

1 -1 0 | 4

0 -1 1 | -2

0 1 -1 | -2

a) So we know no solution possible

b) y = z +2 , x = z + 6

if 0 <=z <=2

2 <=y <=4

6 <=x <=8

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