Two Trains with a TwistDate: 9/11/96 at 22:41:27 From: Anonymous Subject: Two Trains with a Twist The local train heading for Westford at 60km/h and the express train heading for Eastford at 80 km/h pass each other at Centerville while heading in opposite directions along a straight roadbed. Westford and Eastford are 195 km apart, and if the local train makes one additional 15 minute stop after passing Centerville the express completes its run before the local does. What is the maximum distance to the nearest kilometer between Westford and Centerville? Date: 9/14/96 at 21:34:9 From: Doctor Ana Subject: Re: Two Trains with a Twist Westford Centerville Eastford W----------------------------C---------------------------E |----------------------------| x |---------------------------| 195 - x [diagram by Dr. Chuck] First of all, let's set up our variables. Let's call the distance between Westford and Centerville x. This will make the distance between Eastford and Centerville 195-x. We know that distance equals rate times time, or d = rt. This means that time equals distance divided by rate, t = d/r. We can say that it took the express train (195-x)/80 hours to reach Eastford. We can also say that it took the local train x/60 + .25 hours to reach Westford. If the express train got to its destination first, then we can set up the inequality (195-x)/80 < x/60 +.25 This is true because the number of hours to reach Eastford is less than the number of hours to reach Westford. Let's multiply both sides of this inequality by 240 to make it look nicer. 3(195-x) < 4x + 60 585-3x < 4x + 60 Adding 3x to both sides we have: 585 < 7x + 60 Subtracting 60 from both sides we have: 525 < 7x Dividing both sides by 7 we have: 75 < x Remember that x stands for the distance between Centerville and Westford, which is the distance that the local train has to travel after the 2 trains pass at Centerville. This inequality states that x has to be more than 75 kilometers, which should make sense, because the express train goes faster than the local train, so it can go further and still beat the local train. The problem is that you are asked for the maximum distance between Westford and Centerville, and all we have is a minimum distance. First of all, why don't you read the problem again and be sure that they didn't ask for the distance between Eastford and Centerville? If they did ask for the distance between Westford and Centerville, then they could be as much as 195 kilometers apart, because the express train will certainly beat the local train, and that is the only requirement. This is a complicated problem. I hope you understand my explanation and my answer. Try to do this problem again on your own now that you've read how I got my answer and be sure that you read the problem correctly. If you have any more questions please write back. -Doctor Ana, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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