Chapter 5.4, Problem 11E

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

Waiting at a Stop Sign Consider a side road connecting to a major highway at a stop sign. According to a study by D.R. Drew, the average delay $D$, in seconds, for a car waiting at the stop sign to enter the highway is given by

$D=\frac{e^{qt}-1-qt}{q}$,

where $q$ is the flow rate, or the number of cars per second passing the stop sign on the highway, and $T$ is the critical highway that will allow for safe entry. We assume that the critical headway is $T=5\text{seconds}$.

a. What is the average delay time if the flow rate is $500$ cars per hour ( $0.14$ car per second)?

b. The service rate $s$ for a stop sign is the number of cars per second that can leave the stop sign. It is related to the delay by

$s=D^{-1}$.

Use function composition to represent the service rate as a function of flow rate. Reminder: ${\left(a/b\right)}^{-1}=b/a$.

c. What flow rate will permit a stop sign service rate of $5$ cars per minute ( $0.083$ car per second)?