In mathematics, a network is a collection of points (called nodes), with lines (

Question

In mathematics, a network is a collection of points (called nodes), with lines (called branches) connecting some or all of the nodes. The direction of ?ow in each branch is indicated by an arrow on that branch, and the amount of ?ow is either shown or denoted by a variable. We will use network ?ow to model two real-world situations: the ?ow of tra?c in a city and the ?ow of electricity in a circuit.

1. Suppose that we wish to calculate the tra?c ?ow in a system of one-way roads. We can represent such a system schematically as a network, where the nodes are intersections and the branches represent the roads. The direction of ?ow in each road is given by the arrows

The numbers on some of the branches correspond to the number of vehicles per hour passingalong that road. There are ?ve variables, x1; : : : ; x5, corresponding to roads in which thetra?c ?ow is unknown. We wish to use linear algebra to ?nd the tra?c ?ows x1; : : : ; x5.

We can write equations that describe the ?ow through each intersection. For instance, atintersection A the ?ow of tra?c in is 300 + 500 = 800 vehicles per hour, and the ?ow of tra?cout is x1 + x2 vehicles per hour, so the total tra?c ?ow is given by the equation

A : x1 + x2 = 800

(a) Find the equations for the tra?c ?ow at the other intersections B, C and D.

(b) You now have 4 equations in 5 variables. Solve these equations to ?nd the general ?owpattern for the network.

(c) The general solution that you ?nd should contain a parameter, t say. A negative ?ow ina branch of the network corresponds to ?ow opposite to the direction of the arrows. Asall roads are one-way, none of the variables x1, . . . , x5 can be negative. What constraintsdoes this place on the parameter t?

(d) Give a physical interpretation of the parameter t, in terms of the movement of vehicles.

Explanation / Answer

we have to check each node and find the in flow is equal to out flow...

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