Man Crossing a BridgeDate: 09/27/2001 at 16:48:56 From: Eric Yang Subject: Man crossing bridge problem Dear Dr. Math, A man is jogging across a bridge. When he is 3/8 of the way across, he hears a train coming from behind him. He calculates that if he keeps running, he will reach the end of the bridge at the same instant as the train. He also calculates that if he turns around and runs back, he will reach the beginning of the bridge at the same instant as the train. Say the man runs consistently at 8 mph. What is the speed of the train? I tried two solutions and I get 2 mph. Please help. Date: 09/28/2001 at 01:18:10 From: Doctor Ian Subject: Re: Man crossing bridge problem Hi Eric, Here is the man, 3/8 of the way across the bridge, with the train coming from behind, some distance D from the bridge. T M |--|--|--|--|--|--|--|--| |-----| D |-----------------------| B |--------| (3/8)B |--------------| (5/8)B The train will cover distance D in the same time it takes the man to cover distance (3/8)B. The train will cover distance (D+B) in the same time it takes the man to cover distance (5/8)B. Recall that time = distance / rate. We know that (3/8)B D time to run back = ------ = -------------- 8 mph speed of train We also know that (5/8)B D+B time to run forward = ------ = -------------- 8 mph speed of train Now, this looks bad, because we have two equations, but three unknowns. But let's rewrite the second equation to look like this: (3/8)B (2/8)B D B ------ + ------ = -------------- + -------------- 8 mph 8 mph speed of train speed of train This is the first equation, with something extra added to each side. Since the extras must be equal, we know that (2/8)B B ------ = -------------- 8 mph speed of train Do you see why? Note that there is a way to solve the problem without equations, which may be one reason that it comes up so often. Let's say the man chooses to run away from the train. When the train reaches the bridge, he's moved 3/8 of the length of the bridge, meaning he has 2/8, or 1/4, of the bridge left to run. So the man can run 1/4 the length of the bridge in the time it takes the train to move the entire length of the bridge. And we know the man runs at 8 mph. So that tells us the speed of the train, right? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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