Chapter 5.4, Problem 10E

Reminder Round all answers to two decimal places unless otherwise indicated.

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 $q_m$ denote the maximum flow rate, $k$ the mean traffic density (that is, the average number of vehicles per mile), and $k_m$ the density at which flow rate is a maximum (that is, the value of $k$ when $q=q_m$).

a. An important measurement of the traffic on highway is the relative density $R$, which is defined as

$R=\frac{k}{k_m}$.

i. What does a value of $R<1$ indicate about traffic on a highway?

ii. What does a value of $R>1$ indicate about traffic on highway?

b. Let $u$ denote the mean speed of vehicles on the road and $u_f$ the free speed-that is, the speed when there is no traffic congestion at all. One study proposes the following relation between density and speed:

$u=u_f{e}^{-0.5R^2}$.

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 $k_m$ is $122$ cars per mile and $u_f$ is $75$ miles per hour. (include values 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.