Difference between revisions of "1996 AHSME Problems/Problem 13"
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<math> \text{(A)}\ hm\qquad\text{(B)}\ \frac{h}{h+m}\qquad\text{(C)}\ \frac{h}{m-1}\qquad\text{(D)}\ \frac{hm}{m-1}\qquad\text{(E)}\ \frac{h+m}{m-1} </math> | <math> \text{(A)}\ hm\qquad\text{(B)}\ \frac{h}{h+m}\qquad\text{(C)}\ \frac{h}{m-1}\qquad\text{(D)}\ \frac{hm}{m-1}\qquad\text{(E)}\ \frac{h+m}{m-1} </math> | ||
− | ==Solution == | + | __TOC__ |
+ | ==Solution== | ||
+ | If Sunny runs at a rate of <math>s</math> for <math>h</math>. Then the distance covered is <math>sh</math>. Now we know that Moonbeam runs <math>m</math> times as fast than Sunny, so Moonbeam runs at the rate of <math>ms</math>. Now Moonbeam gave Sunny a headstart of <math>h</math> meters, so he will catch on Sunny at the rate of <math>s(m-1)</math> . At time <math>\frac{h}{m-1}</math> Moon beam will catch on Sunny. Now we are asked how much in meters he have to run to catch on Sunny. That is <math>\frac{hm}{m-1}</math>. | ||
− | + | ===Solution 2=== | |
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− | ==Solution 2== | ||
Note that <math>h</math> is a length, while <math>m</math> is a dimensionless constant. Thus, <math>h</math> and <math>m</math> cannot be added, and <math>B</math> and <math>E</math> are not proper answers, since they both contain <math>h+m</math>. | Note that <math>h</math> is a length, while <math>m</math> is a dimensionless constant. Thus, <math>h</math> and <math>m</math> cannot be added, and <math>B</math> and <math>E</math> are not proper answers, since they both contain <math>h+m</math>. | ||
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If <math>m</math> is a very, very large number, then Moonbeam will have to run just over <math>h</math> meters to reach Sunny. Or, in the language of limits: | If <math>m</math> is a very, very large number, then Moonbeam will have to run just over <math>h</math> meters to reach Sunny. Or, in the language of limits: | ||
− | <math>\lim_{ | + | <math>\lim_{m\rightarrow \infty} d(m) = h</math>, where <math>d(m)</math> is the distance Moonbeam needs to catch Sunny at the given rate ratio of <math>m</math>. |
In option <math>A</math>, when <math>m</math> gets large, the distance gets large. Thus, <math>A</math> is not a valid answer. | In option <math>A</math>, when <math>m</math> gets large, the distance gets large. Thus, <math>A</math> is not a valid answer. | ||
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In option <math>D</math>, when <math>m</math> gets large, the ratio <math>\frac{m}{m-1}</math> gets very close to, but remains just a tiny bit over, the number <math>1</math>. Thus, when you multiply it by <math>h</math>, the ratio in option <math>D</math> gets very close to, but remains just a tiny bit over, <math>h</math>. Thus, the best option out of all the choices is <math>\boxed{D}</math>. | In option <math>D</math>, when <math>m</math> gets large, the ratio <math>\frac{m}{m-1}</math> gets very close to, but remains just a tiny bit over, the number <math>1</math>. Thus, when you multiply it by <math>h</math>, the ratio in option <math>D</math> gets very close to, but remains just a tiny bit over, <math>h</math>. Thus, the best option out of all the choices is <math>\boxed{D}</math>. | ||
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+ | ===Solution 3=== | ||
+ | |||
+ | Assume that Sunny originally runs at a unit speed, and thus Moonbeam runs at a rate of <math>m</math>. | ||
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+ | Choose a new reference frame where Sunny is still, and Moonbeam runs at a rate of <math>m-1</math>. In this new reference frame, the distance to be run is still <math>h</math>. | ||
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+ | Moonbeam runs this distance <math>h</math> in a time of <math>\frac{h}{m-1}</math> | ||
+ | |||
+ | Returning to the original reference frame, if Moonbeam runs for <math>\frac{h}{m-1}</math> seconds, Moonbeam will cover a distance of <math>\frac{hm}{m-1}</math>, which is option <math>\boxed{D}</math>. | ||
==See also== | ==See also== | ||
{{AHSME box|year=1996|num-b=12|num-a=14}} | {{AHSME box|year=1996|num-b=12|num-a=14}} | ||
+ | |||
+ | [[Category:Introductory Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 19:34, 2 November 2023
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?
Contents
[hide]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. Now 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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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