1996 AHSME Problems/Problem 13
Problem
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?
Solution
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
.
Solution 2
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
.
Solution 3
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
.
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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