Man and Train on a Bridge
Date: 2/1/96 at 16:27:56 From: Anonymous Subject: Story problem A man is walking across a railroad bridge that goes from point A to point B. He starts at point A, and when he is 3/8 of the way across the bridge, he hears a train approaching. The train's speed is 60 mph (miles per hour). The man can run fast enough so that if he turns and runs back toward point A, he will meet the train at A, and if he runs forward toward point B, the train will overtake him at B. How fast can the man run? I have no idea how to do this problem.
Date: 3/24/96 at 22:54:16 From: Doctor Rachel Subject: Re: Story problem Hi! If the man turns and runs toward point A, he will cover 3/8 of the length of the bridge in the time that it takes the train to reach A. If the man runs forward toward point B, what part of the bridge will he cover before the train reaches A? Well, he will cover 3/8 of the bridge, only heading forward toward B. This will put him 3/8 + 3/8 = 6/8 = 3/4 of the way across the bridge by the time the train reaches A. Since we know that the man and the train will meet at B, this means that in the time it takes the man to run the remaining 1/4 of the bridge, the train will cover the entire length of the bridge. If it takes the man the same time to cover 1/4 of the bridge that it takes the train to cover the whole bridge, then the train must be going four times as fast as the man. Another way of saying this is that the man runs at 1/4 the speed of the train. Since the train's speed is known to be 60 mph, this means that the man runs at (1/4) 60 = 15 mph. _____________________________________________________________ An equation-oriented solution to this problem makes use of the distance formula, which tells us that time = distance/rate. We will use this to express the information that we know about the train and the man when the man is 3/4 of the way across the bridge and the train is at A. The period of time that we are considering is the same for both the train and the man. During this time, however, they travel different distances. If the bridge has length d, the train travels d while the man only travels 1/4 d. The man's rate is unknown (it's what we want to figure out) so we will call it r. The train's rate is 60 mph. This information makes it possible to write the two following equations: Time it takes the man to run 1/4 the bridge length = (1/4 d)/r Time it takes the train to cover the entire bridge = d/60 We can set these two equations equal to each other because the two times are equal. This produces the following: 1/4 d d ----- = --- r 60 Cross-multiplying results in: 60 (1/4 d) = dr We can divide both sides of the equation by d because d cannot possibly be zero (the bridge has a real length): 60 (1/4) = r which leaves us with r = 15 mph, the same answer as above. -Doctors Judy and Rachel, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 8/1/97 at 12:00:00 From: Anonymous Subject: Man walking across bridge. I read with interest your solution to FSarre6399@aol.com 2/1/96 Man walking across bridge problem. I think my solution is simpler and more elegant than the one you gave! Solution: Let M stand for the man's speed in mph. When the man runs toward point A, the relative speed of the train with respect to the man is the train's speed plus the man's speed (60 + M). When he runs toward point B, the relative speed of the train is the train's speed minus the man's speed (60 - M). When he runs toward the train the distance he covers is 3 units. When he runs in the direction of the train the distance he covers is 5 units. We can now write that the ratio of the relative speed of the train when he is running toward point A to the relative speed of the train when he is running toward point B, is equal to the inverse ratio of the two distance units or (60 + M) 5 ----------- = --- (60 - M) 3 3(60 + M) = 5(60 - M) 3*60 + 3M = 5*60 - 5M 8M = 120 M = 15 mph
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