Linear Algebra

Topic: Analyzing Networks 77
We can use Kirchhoff’s Law, that the flow into any intersection equals the flow
out, to establish some equations modeling how traffic flows work here.
(a) Label each of the three arcs of road in the circle with a variable. For each of
the three in-out intersections, get an equation describing the traffic flow at that
node.
(b) Solve that system.
3 This is a map of a network of streets. Below we will describe the flow of cars
into, and out of, this network.
Willow
Shelburne St
Jay Ln
west Winooski Ave
The hourly flow of cars into this network’s entrances, and out of its exits can be
observed.
east Winooski west Winooski Willow Jay Shelburne
into 100 150 25 – 200
out of 125 150 50 25 125
(The total in must approximately equal the total out over a long period of time.)
Once inside the network, the traffic may proceed in different ways, perhaps
filling Willow and leaving Jay mostly empty, or perhaps flowing in some other
way. We can use Kirchhoff’s Law that the flow into any intersection equals the
flow out.
(a) Determine the restrictions on the flow inside this network of streets by setting
up a variable for each block, establishing the equations, and solving them. Notice
that some streets are one-way only. (Hint: this will not yield a unique solution,
since traffic can flow through this network in various ways. You should get at
least one free variable.)
(b) Suppose some construction is proposed for Winooski Avenue East between
Willow and Jay, so traffic on that block will be reduced. What is the least
amount of traffic flow that can be allowed on that block without disrupting the
hourly flow into and out of the network?
4 Calculate the amperages in this network with more than one voltage rise.
1.5 volt
5ohm
2ohm 3volt
10 ohm
east
3ohm
6ohm
5 In the circuit with the 8 ohm and 12 ohm resistors in parallel, the electric current
away from and back to the battery was found to be 25/6 amperes. Thus, the