Chapter 4.1, Problem 19E

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

Headway on Four-Lane Highways When traffic is flowing on a highway, the headway is the average time between vehicles. On four-lane highways, the probability $P$ that the headway is at least $t$ seconds is given to a good degree of accuracy by

$P=e^{-qt}$,

where $q$ is the average number of vehicles per second traveling one way on the highway.

a	On a four-lane highway carrying an average of $500$ vehicles per hour in one direction, what is the probability that the headway is at least $15\text{ seconds}$? (Note: $500$ vehicles per hour is $500/3600=0.14$ vehicle per second.)
b	On a four-lane highway carrying an average of $500$ vehicles per hour, what is the decay factor for the probability that headways are at least $t\text{ seconds}$? Reminder: An important law of exponent tells us that $a^{bt}={\left(a^b\right)}^t$.