Automobile traffic passes a point P on a road of width w feet at an average rate of R vehicles per second.
Although the arrival of automobiles is irregular, traffic engineers have found that the average waiting time T until there is a gap in traffic of at least t seconds is approximately T = te^Rt seconds. A pedestrian walking at a speed of 3.5 ft/s (5.1 miles per hour) requires t = w/3.5 s to cross the road. Therefore, the average time the pedestrian will have to wait before crossing is f (w, R) = (w/3.5)e^wR/3.5 s.

(a) What is the pedestrian’s average waiting time if w = 25 ft and R = 0.2 vehicle per second?

(b) Use the Linear Approximation to estimate the increase in waiting time if w is increased to 27 ft.

(c) Estimate the waiting time if the width is increased to 27 ft and R decreases to 0.18.

(d) What is the rate of increase in waiting time per 1-ft increase in width when w = 30 ft and R = 0.3 vehicle per second?