In this workshop, we will consider the following traffic network consisting of one-way streets: a3 a2 a4 2 a1 a5 4 a6 a7 The network consists of four junctions (coloured in blue), five "internal" streets between the junctions (labelled x1 to x5), and eight "external" streets that connects this network to the rest of the traffic grid (labelled a1 to ag). The direction of traffic on each street is indicated by the red arrow on the street. Suppose traffic counters are placed on the eight external streets, and the average numbers of cars per hours (travelling in the indicated direction) are found to be a1 = 40, a2 = 5, a3 = 15, a4 = 25, a5 = 30, a6 = 15, a7 = 10, as = 20 Using these measurements, we would like to say something about the traffic on the internal streets. Assume that the cars are always in motion in the network, i.e., there are no cars entering/leaving garages on the street, or stopping to park on the street. 2T . By considering the traffic at each junction, write down four equations involving z, to 5. [2 marks] . Without further computation, explain why the eight traffic counters are not sufficent to determine the traffic flow on all streets. [1 marks] Without further computation, how many extra traffic counters do you expect you need to determine the traffic flow on all streets? Briefly justify vour answer. [1 marks] Write down the system of equations in matrix form, then compute the reduced row echelon form of the augmented matrix. You should do this by hand and submit your working, but you may (and are highly recommended to) verify your solution using CAS or Matlab. [4 marks] . Hence write down the solution for the average mumber of cars per hour on each street (you may have parameters in your solution). [2 marks] How many extra traffic counters do you need to completely determine all the traffic low in the networks? Where should you place the extra counters? Make sure yvou explain and justify your answers. [2 marks] Your solution should have some parameters. Give an explanation for what the vectors associated with these parameters represent. To answer this, you may find it useful to draw the road network and add the traffic on each road for different values of the parameters (varying them separately). [2 marks] Not every solution in the full solution space corresponds to a realistic traffic flow. In particular we require that x; ...z to be all non-negative. What is the minimum nmumber of cars per hour on the street x4 in real life? [2 marks] In real life, what is the minimum total traffic possible in the network? In other words, what is the minimum of x7 + 22 + 3 + 4 + 257 Find the traffic low when the minimum is realized and sketch the network labelling the traffic on all the streets with direction. [2 marks]