Course:MATH110/Archive/2010-2011/003/Notes/Optimization/Problem 8

Question

While driving from British Columbia to Alberta, you have to drive through the Mount Macdonald tunnel which is 14.7km long (this makes it the longest tunnel of the Americas by the way).

If we assume that cars drive through the tunnel at a constant speed of $v$ kilometres per hour; that each car is 4 metres long; and that each car respects a safety distance from each other of $0.004v^{2}$ (this means we assume that the distance between cars is proportional to the square of the speed, which itself is roughly the stopping distance, so a more than realistic assumption).

Then what is the constant speed $v$ which ensure the maximal debit of cars through the tunnel? (We define the debit of cars as the number of cars going through the tunnel per hour).