Traffic flow: For traffic moving along a highway, we use q to denote the mean flow rate. That is the average number of vehicles per hour passing a certain point. We let qm denote the maximum flowrate, k the mean traffic density (that is, the average number of vehicles per mile), and km the density at which flow rate is a maximum (that is, the value of k when q = qm).
a. An important measurement of traffic on a highway is the relative density R, which is defined as
i. What does a value of R <>
ii. What does a value of R > 1 indicate about traffic on a highway?
b. Let u denote the mean speed of vehicles on the road and uf the free speed—that is, the speed when there is no traffic congestion at all. One study43 proposes the following relation between density and speed:
Use function composition to find a formula that directly relates mean speed to mean traffic density.
c. Make a graph of mean speed versus mean traffic density, assuming that km is 122 cars per mile and uf is 75 miles per hour. (Include values of mean traffic density up to 250 vehicles per mile.) Paying particular attention to concavity, explain the significance of the point k = 122 on the graph.
d. Traffic is considered to be seriously congested if the mean speed drops to 35 miles per hour. Use the graph from part c to determine what density will result in serious congestion.