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Difference between revisions of "2001 AMC 8 Problems/Problem 19"

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==Problem==
 
==Problem==
  
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Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the speed for the same distance. If Car N's speed and time are shown as solid line, which graph illustrates this?
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Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the speed for the same distance. If Car M and Car N's speed and time are shown as solid line, which graph illustrates this?
  
 
<asy>
 
<asy>
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</asy>
 
</asy>
  
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==Solution==
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==Solution 1 ==
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Since car N has twice the speed, it must be twice as high on the speed axis. Also, since cars M and N travel at the same distance but car N has twice the speed, car N must take twice the time. Therefore, line N must be half the size of line M. Since the speeds are constant, both lines are horizontal. Reviewing the graphs, we see that the only one satisfying these conditions is graph <math> \boxed{\text{D}} </math>.
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Since car N has twice the speed, it must be twice as high on the speed axis. Also, since cars M and N travel at the same distance but car N has twice the speed, car N must take half the time. Therefore, line N must be half the size of line M. Since the speeds are constant, both lines are horizontal. Reviewing the graphs, we see that the only one satisfying these conditions is graph <math> \boxed{\text{D}} </math>.
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==Solution 2 ==
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Since Car N goes at a faster speed than Car M, it will take less time to get to the same destination. Reviewing the graphs, we see that only one graph fits this criteria, graph <math> \boxed{\text{D}} </math>
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2001|num-b=18|num-a=20}}
 
{{AMC8 box|year=2001|num-b=18|num-a=20}}
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{{MAA Notice}}

Latest revision as of 22:05, 13 August 2019

Problem

Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the speed for the same distance. If Car M and Car N's speed and time are shown as solid line, which graph illustrates this?

[asy] unitsize(12); draw((0,9)--(0,0)--(9,0)); label("time",(4.5,0),S); label("s",(0,7),W); label("p",(0,6),W); label("e",(0,5),W); label("e",(0,4),W); label("d",(0,3),W); label("(A)",(-1,9),NW); draw((0,4)--(4,4),dashed); label("M",(4,4),E); draw((0,8)--(4,8),linewidth(1)); label("N",(4,8),E); draw((15,9)--(15,0)--(24,0)); label("time",(19.5,0),S); label("s",(15,7),W); label("p",(15,6),W); label("e",(15,5),W); label("e",(15,4),W); label("d",(15,3),W); label("(B)",(14,9),NW); draw((15,4)--(19,4),dashed); label("M",(19,4),E); draw((15,8)--(23,8),linewidth(1)); label("N",(23,8),E); draw((30,9)--(30,0)--(39,0)); label("time",(34.5,0),S); label("s",(30,7),W); label("p",(30,6),W); label("e",(30,5),W); label("e",(30,4),W); label("d",(30,3),W); label("(C)",(29,9),NW); draw((30,4)--(34,4),dashed); label("M",(34,4),E); draw((30,2)--(34,2),linewidth(1)); label("N",(34,2),E); draw((0,-6)--(0,-15)--(9,-15)); label("time",(4.5,-15),S); label("s",(0,-8),W); label("p",(0,-9),W); label("e",(0,-10),W); label("e",(0,-11),W); label("d",(0,-12),W); label("(D)",(-1,-6),NW); draw((0,-11)--(4,-11),dashed); label("M",(4,-11),E); draw((0,-7)--(2,-7),linewidth(1)); label("N",(2,-7),E); draw((15,-6)--(15,-15)--(24,-15)); label("time",(19.5,-15),S); label("s",(15,-8),W); label("p",(15,-9),W); label("e",(15,-10),W); label("e",(15,-11),W); label("d",(15,-12),W); label("(E)",(14,-6),NW); draw((15,-11)--(19,-11),dashed); label("M",(19,-11),E); draw((15,-13)--(23,-13),linewidth(1)); label("N",(23,-13),E); [/asy]

Solution 1

Since car N has twice the speed, it must be twice as high on the speed axis. Also, since cars M and N travel at the same distance but car N has twice the speed, car N must take half the time. Therefore, line N must be half the size of line M. Since the speeds are constant, both lines are horizontal. Reviewing the graphs, we see that the only one satisfying these conditions is graph $\boxed{\text{D}}$.


Solution 2

Since Car N goes at a faster speed than Car M, it will take less time to get to the same destination. Reviewing the graphs, we see that only one graph fits this criteria, graph $\boxed{\text{D}}$

See Also

2001 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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
All AJHSME/AMC 8 Problems and Solutions

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