Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2019 AMC 8 Problems/Problem 5"

(Blanked the page)
(Tag: Blanking)
m (→‎Solution 1 - Using the answers)
 
(32 intermediate revisions by 23 users not shown)
Line 1: Line 1:
 +
==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?
  
 +
[[File:2019_AMC_8_-4_Image_1.png|900px]]
 +
 +
[[File:2019_AMC_8_-4_Image_2.png|600px]]
 +
 +
==Solutions==
 +
 +
 +
==Solution 1 - Using the answers==
 +
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>.
 +
Third, the tortoise wins the race, meaning that the non-constant one should go off the graph last, ruling out <math>(A)</math>.
 +
Therefore, the answer <math>\boxed{\textbf{(B)}}</math> is the only one left.
 +
 +
<math>\phantom{Note to the original author of this solution: "we shouldn't be able to edit" is incorrect (if its definition is what I think it is), because I was able to edit. Also, I deleted that, (but did nothing else) }</math>
 +
 +
==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
 +
 +
==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 ==
 +
 +
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==
 +
https://youtu.be/FDG99LC75vA
 +
 +
~savannahsolver
 +
 +
==Video Solution by The Power of Logic(1 to 25 Full Solution)==
 +
https://youtu.be/Xm4ZGND9WoY
 +
 +
~Hayabusa1
 +
 +
==See also==
 +
{{AMC8 box|year=2019|num-b=4|num-a=6}}
 +
 +
{{MAA Notice}}

Latest revision as of 00:58, 22 November 2024

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 $d$ traveled by the two animals over time $t$ from start to finish?

2019 AMC 8 -4 Image 1.png

2019 AMC 8 -4 Image 2.png

Solutions

Solution 1 - Using the answers

First, the tortoise walks at a constant rate, ruling out $(D)$ Second, when the hare is resting, the distance will stay the same, ruling out $(E)$ and $(C)$. Third, the tortoise wins the race, meaning that the non-constant one should go off the graph last, ruling out $(A)$. Therefore, the answer $\boxed{\textbf{(B)}}$ is the only one left.

$\phantom{Note to the original author of this solution: "we shouldn't be able to edit" is incorrect (if its definition is what I think it is), because I was able to edit. Also, I deleted that, (but did nothing else) }$

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 $\boxed{\textbf{(B)}}$. 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

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

https://youtu.be/FDG99LC75vA

~savannahsolver

Video Solution by The Power of Logic(1 to 25 Full Solution)

https://youtu.be/Xm4ZGND9WoY

~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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_5&oldid=235268"