Chapter 5.4, Problem 12E

Average Traffic Spacing The headway $h$ is the average time between vehicles. On a highway carrying an average of $500$ vehicles per flour, the probability $P$ that the headway is at least $t$ seconds is given by

$P={0.87}^t$.

a. What is the limiting value of $P$? Explain what this means in practical terms.

b. The headway $h$ can be calculated as the quotient of the spacing $f$, in feet, which is the average distance between vehicles, and the average speed $v$, in feet per second, of traffic. Thus, the probability that spacing is at least $f$ feet is the same as the probability that the headway is at least $f/v$ seconds. Use function composition to find a formula for the probability $Q$ that the spacing is at least $f$ feet. (Note: Your formula will involve both $f$ and $v$).

c. If the average speed is $88$ feet per second ( $60$ miles per hour), what is the probability that the spacing between two vehicles is at least $40$ feet?