Can someone tell me if I did this problem correctly?
Your task is to show how you can use systems of linear equations to study the flow of traffic through a traffic system. The traffic system consists of streets and intersections where the streets meet. Each street has a certain flow, which is measured in vehicles per unit of time. We will calculate the traffic flow for all streets in the traffic system, given the flow into the traffic system. We make an important assumption: Traffic flow is conserved at each intersection. This means that a vehicle reaching an intersection must continue through the traffic system. A positive value on the flow flows in the direction of the arrow, and a negative flow flows in the opposite direction.
Example (not related to the figure):
If the flow into a four-way intersection is y₁ vehicles/time unit and 30 vehicles/time unit respectively and the flow out of the intersection is y₂ vehicles/time unit and 50 vehicles/time unit respectively, then it applies that y₁+30=y₂+50 at the intersection. After simplification, we get y₁-y₂=20. Each crossing thus gives a linear equation in unknown flows y₁ and y₂. If you solve the system of equations, you can determine the flow through the traffic system.
Q:
a) Is our flow conservation assumption valid?
b) When can the assumption be incorrect?
c) What other simplifications have we adopted?
Study the road system in Figure 1. The numbers indicate the traffic flows in vehicles per hour. All roads are one-way and traffic follows the direction of the arrows.
d) Set up the system of equations
e) Determine the traffic flow for the road system.
f) If the traffic flow for one of the roads x, y, z or w<= 100 vehicles per hour due to road work. What will be the traffic flow for the other roads?
a) It's a valid assumption. This assumption holds that the traffic flow passes through each intersection and continues in one of the four directions. There is no other place for the traffic to go but through the intersection.
b) The assumption is incorrect when traffic flow is not preserved at the intersection. This may be the case, for example, if there is a traffic light that regulates the flow at the intersection. The traffic light may prevent vehicles reaching the intersection from continuing through the traffic system. But, some would argue that they wouldn't necessarily say that a traffic light breaks the assumption. Once the light turns green the traffic must still proceed through the intersection.
I think that the assumption fails at the endpoints of the study because when we stop collecting data there are likely cars sitting at the intersection which have arrived but not yet left. Another time that the assumption fails is if there are accidents in an intersection where a car arrives but never leaves in one piece.
c) We have assumed that the traffic flow is constant along the streets, and that there are no flows between the streets. This is realistic if the streets are long and/or if the traffic is low.
d) From the description I expected variables:
w: N/S flow in to C and N/S flow out of B
x: E/W flow in to A and E/W flow out of B
y: S/N flow in to A and S/N flow out of D
z: W/E flow in to C and W/E flow out of D
We would then get one equation for each intersection. For instance, the equation for intersection A would be x+y=190+241
For (e), if I'm understanding the problem correctly, I don't think you can get a unique solution.
Question (f) sort of implies that there isn't a unique solution because if there was only one possible solution then you couldn't say that the traffic flow for one of the roads changes suddently.
I think question (f) is asking you to interpret your solution in (e) and wants something like: If flow on w is stopped by roadwork then flow on x and z will have to increase. If you look at the diagram, you could see that much without doing any math. If there is suddenly no flow on w, then all of the traffic coming into B has to leave via x, and all of the traffic leaving at C has to have come from z. However, there's still work to be done to figure out what effect roadwork on w would have on street y if any.