Welcome to a competition that was held in 1996 by the UNIVERSITY OF TORONTO MATHEMATICS NETWORK. Although that competition is over, you are still invited to try your hand at exploring mathematics in action in this real life situation!Keep The Traffic Moving!
In this competition, you are the traffic engineer. You control the timings of the lights at the six intersections shown in the picture below. Each road has two lanes, one in each direction. The lanes are each 4 metres wide (so each road is 8 metres wide), and the spacing between the roads is as shown. For convenience we've numbered the lanes from 1 to 10, indicated their directions, and labelled the lights A through F.
Lane Lane Lane Lane Lane Lane /\ 5 | | | 6 7 | | | 8 9 | | |10 /\ || ||| | |/\ ||| | |/\ ||| | |/\ || 20 m North <- \/| | ||| \/| | ||| \/| | ||| \/ Lane 1 -------- -------- -------- -------- -------- A -------- B -------- C -------- Lane 2 -------- -------- -------- -------- -> | | | | | | | | | /\ | | | | | | | | | || 32 m <- | | | | | | | | | \/ Lane 3 -------- -------- -------- -------- -------- D -------- E -------- F -------- Lane 4 -------- -------- -------- -------- -> | | | | | | | | | /\ | | | | | | | | | || 20 m <------>| | |<------>| | |<------>| | |<------> \/ 72 m 72 m 72 m 72 mYou can control these lights yourself through a computer simulation available on our web site.
You have also found that the traffic behaviour (the way a car speeds up or slows down when faced with an obstacle in front, such as a red light or another car) follows a certain mathematical model governed by three numbers c, l, and L (the model is described in the section Details on the Mathematical Model Used).
To get the full details on each part you can use the links at the top and bottom of each page. Here's a brief description of them:
In the detailed discussion of this part, we'll give you hints on how basic algebra can be used to come up with a simple set of inequalities that tells whether or not traffic lights would be capable of handling the traffic flow. See if you can figure out this formula. You can test your answer on the computer simulation.
In the detailed discussion of this part, we'll explain how ideas of calculus and " differential equations" can be used to answer the question, and challenge you to find out how long it takes for the line to get moving even if everybody's reaction time is instantaneous and there's no limit on the cars' acceleration power. You may be surprised at the answer, and less inclined to blame the cars in front of you next time you're kept waiting! Note: the mathematics in this part is quite challenging and requires a knowledge of calculus. But you can still watch what happens on the computer simulation even if you can't solve the problem mathematically.
Use the links above or below to get more information on each of these three competition questions and on the mathematical model used, or to access the computer simulation.
Good luck!