Difference between revisions of "2019 AMC 8 Problems/Problem 5"
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| − | == Problem | + | ==Problem== |
A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance <math>d</math> traveled by the two animals over time <math>t</math> from start to finish? | A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance <math>d</math> traveled by the two animals over time <math>t</math> from start to finish? | ||
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[[File:2019_AMC_8_-4_Image_2.png|600px]] | [[File:2019_AMC_8_-4_Image_2.png|600px]] | ||
| − | ==Solution 1== | + | ==Solutions== |
| + | |||
| + | |||
| + | ==Solution 1 - Using the answers== | ||
First, the tortoise walks at a constant rate, ruling out <math>(D)</math> | First, the tortoise walks at a constant rate, ruling out <math>(D)</math> | ||
Second, when the hare is resting, the distance will stay the same, ruling out <math>(E)</math> and <math>(C)</math>. | Second, when the hare is resting, the distance will stay the same, ruling out <math>(E)</math> and <math>(C)</math>. | ||
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==Solution 2== | ==Solution 2== | ||
| − | First, we know that the rabbit beats the tortoise in the first half of the race. So he is going to be ahead of the tortoise. We also know, while he rested, he didn't move. The only graph portraying that is going to be <math>\boxed{\textbf{(B)}}</math>. This is our answer. Just remember, since the tortoise finished faster than the hare, his line will be a lesser value on the time-axis. ~bobthefam, Edited by Rishi09 | + | First, we know that the rabbit beats the tortoise in the first half of the race. So he is going to be ahead of the tortoise. We also know, while he rested, he didn't move. The only graph portraying that is going to be <math>\boxed{\textbf{(B)}}</math>. This is our answer. Just remember, since the tortoise finished faster than the hare, his line will be a lesser value on the time-axis. |
| + | ~bobthefam, Edited by Rishi09 | ||
| + | |||
| + | ==Video Solution by Math-X (First fully understand the problem!!!)== | ||
| + | https://youtu.be/IgpayYB48C4?si=zrQW67K2J_vVISHd&t=871 | ||
| + | |||
| + | ~Math-X | ||
| + | |||
| + | ==Video Solution (HOW TO THINK CREATIVELY!!!)== | ||
| + | https://youtu.be/AH2uNkg67oo | ||
| + | |||
| + | ~Education, the Study of Everything | ||
== Video Solution == | == Video Solution == | ||
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~savannahsolver | ~savannahsolver | ||
| + | |||
| + | ==Video Solution by The Power of Logic(1 to 25 Full Solution)== | ||
| + | https://youtu.be/Xm4ZGND9WoY | ||
| + | |||
| + | ~Hayabusa1 | ||
==See also== | ==See also== | ||
Latest revision as of 00:58, 22 November 2024
Contents
- 1 Problem
- 2 Solutions
- 3 Solution 1 - Using the answers
- 4 Solution 2
- 5 Video Solution by Math-X (First fully understand the problem!!!)
- 6 Video Solution (HOW TO THINK CREATIVELY!!!)
- 7 Video Solution
- 8 Video Solution 2
- 9 Video Solution 3
- 10 Video Solution by The Power of Logic(1 to 25 Full Solution)
- 11 See also
Problem
A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance 
traveled by the two animals over time 
from start to finish?
Solutions
Solution 1 - Using the answers
First, the tortoise walks at a constant rate, ruling out 
Second, when the hare is resting, the distance will stay the same, ruling out 
and 
.
Third, the tortoise wins the race, meaning that the non-constant one should go off the graph last, ruling out 
.
Therefore, the answer 
is the only one left.
Solution 2
First, we know that the rabbit beats the tortoise in the first half of the race. So he is going to be ahead of the tortoise. We also know, while he rested, he didn't move. The only graph portraying that is going to be . This is our answer. Just remember, since the tortoise finished faster than the hare, his line will be a lesser value on the time-axis.
~bobthefam, Edited by Rishi09
Video Solution by Math-X (First fully understand the problem!!!)
https://youtu.be/IgpayYB48C4?si=zrQW67K2J_vVISHd&t=871
~Math-X
Video Solution (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
Video Solution
The Learning Royal: https://youtu.be/IiFFDDITE6Q
Video Solution 2
Solution detailing how to solve the problem: https://www.youtube.com/watch?v=uQTM8Kmh9B8&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=6
Video Solution 3
~savannahsolver
Video Solution by The Power of Logic(1 to 25 Full Solution)
~Hayabusa1
See also
| 2019 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
