1996 AHSME Problems/Problem 13
Sunny runs at a steady rate, and Moonbeam runs times as fast, where is a number greater than 1. If Moonbeam gives Sunny a head start of meters, how many meters must Moonbeam run to overtake Sunny?
If Sunny runs at a rate of for . Then the distance covered is . Now we know that Moonbeam runs times as fast than sunny, so Moonbeam runs at the rate of . Now moonbeam gave sunny a headstart of meters, so he will catch on Sunny at the rate of . At time Moon beam will catch on Sunny.N ow we are asked how much in meters he have to run to catch on Sunny.That is .
Note that is a length, while is a dimensionless constant. Thus, and cannot be added, and and are not proper answers, since they both contain .
Thus, we only concern ourselves with answers .
If is a very, very large number, then Moonbeam will have to run just over meters to reach Sunny. Or, in the language of limits:
, where is the distance Moonbeam needs to catch Sunny at the given rate ratio of .
In option , when gets large, the distance gets large. Thus, is not a valid answer.
In option , when gets large, the distance approaches , not as desired. This is not a valid answer. (In fact, this is the distance Sunny runs, which does approach as Moonbeam gets faster and faster.)
In option , when gets large, the ratio gets very close to, but remains just a tiny bit over, the number . Thus, when you multiply it by , the ratio in option gets very close to, but remains just a tiny bit over, . Thus, the best option out of all the choices is .
Assume that Sunny originally runs at a unit speed, and thus Moonbeam runs at a rate of .
Choose a new reference frame where Sunny is still, and Moonbeam runs at a rate of . In this new reference frame, the distance to be run is still .
Moonbeam runs this distance in a time of
Returning to the original reference frame, if Moonbeam runs for seconds, Moonbeam will cover a distance of , which is option .
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