Hines Parkway Problem

Date: 4 Feb 1995 15:02:03 -0500
From: Anonymous
Subject: math problem

The Hines Parkway is a 20 mile road along a river that has no stop lights.
The speed limit is 40 mph; but cars typically travel at 45-50 mph or perhaps
faster.  During traffic it is hard to pass, and traffic backs up behind any
car that is going slower than the "average" speed.

What is the average speed of cars on Hines Parkway given the following
information?  A car enters one end of Hines Parkway and drives at 40 mph.
Other cars begin to arrive behind him, and because they are traveling 
faster than 40 mph, a line of cars traveling 40 mph begins to form.  If 
after seven miles there is a line of 14 cars, what is the average speed of 
cars before they must slow down behind the line of cars?  There were only 
two cars that entered Hines Park at the same time.  A sheriff regularly 
patrols the parkway, so speeds in excess of 50 mph are rare.  Is there 
enough information to answer this question?  If not, what other information 
is needed.

Thanks, 
Terry Shannon
The Sheriff <g>.   Just kidding!  I just wanted to know if lines of cars
forming on roads can give any indication of the speeds of arriving cars.

Date: 5 Feb 1995 00:11:26 GMT
From: Dr. Math
Subject: Re: math problem

Hello!

Well, I don't think there's enough information to solve this problem.  See,
the number of cars that pile up behind Driver A isn't just a function of
how fast they are going, it's also a function of how they're spaced along
the parkway.  So if seven cars arrive shortly after Car A, and they all
travel at 41 mph, they might all pile up behind Car A.  But if they arrive
twenty minutes after Car A and they all travel at 41 mph, it's likely that
none of them will pile up behind Car A.  So it's heavily dependent on how
those cars are spaced.

The information you'd need in order to make any conclusions is the time
that the cars in the pile-up entered the road.  Then you'd be able to look
at the last car in the pile, and get a lower bound on how fast its driver
was going (it'd have to be going at least a certain speed if it wanted to
make up the given distance in the given time).  

Given your information, all you could conclude is that seven cars had
gone faster than 40 mph.

-Ken "Dr." Math