Difference between revisions of "2022 AMC 8 Problems/Problem 10"
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MRENTHUSIASM (talk | contribs) (→Solution 3: Elimination Again (But with more math)) |
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− | ==Solution== | + | ==Solution 1 (Analysis)== |
Note that: | Note that: | ||
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~MRENTHUSIASM | ~MRENTHUSIASM | ||
+ | |||
+ | ==Solution 2 (Elimination)== | ||
+ | Ling's trip took <math>2</math> hours, thus she traveled for <math>2 \times 45=90</math> miles. Choices <math>\textbf{(B)}</math>, <math>\textbf{(C)}</math>, and <math>\textbf{(D)}</math> are eliminated. | ||
+ | Ling drove <math>45</math> miles per hour (mph) to the mountains, and <math>60</math> mph back to her house, so the rightmost slope must be steeper than the leftmost one. Choice <math>\textbf {(A)}</math> is eliminated. | ||
+ | This leaves us with <math>\boxed{\textbf{(E)}}</math>. | ||
+ | |||
+ | ==Solution 3 (Elimination)== | ||
+ | |||
+ | Using the <math>\text{speed} = \frac{\text{distance}}{\text{time}}</math> formula, and plugging in the values <math>45</math> mph and <math>2</math> hrs, we get that the distance from Ling's house to the mountains is <math>90</math> miles. That means the first slope ends at <math>90,</math> so choices <math>\textbf{(B)}, \textbf{(C)},</math> and <math>\textbf{(D)}</math> are eliminated. | ||
+ | |||
+ | Furthermore, if the distance is <math>90</math> miles, and Ling is returning home at <math>60</math> mph, it must have taken her <math>1.5</math> hours. Adding <math>1.5</math> and <math>3</math> (how long she hiked for) to her arrival time, <math>10</math> AM, we see she must have come back home at <math>2:30</math> PM. Choice <math>\textbf{(A)}</math> is eliminated, so the only valid choice left is choice <math>\boxed{\textbf{(E)}}.</math> | ||
+ | |||
+ | ~ProProtractor | ||
+ | |||
+ | ==Video Solution by Math-X (First understand the problem!!!)== | ||
+ | https://youtu.be/oUEa7AjMF2A?si=GriQR9m9WenwQVdf&t=1311 | ||
+ | |||
+ | ~Math-X | ||
+ | |||
+ | ==Video Solution (CRITICAL THINKING!!!)== | ||
+ | https://youtu.be/Q3G-qyCUnYI | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | |||
==Video Solution== | ==Video Solution== | ||
− | https:// | + | https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=733 |
~Interstigation | ~Interstigation | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/1xspUFoKDnU?t=332 | ||
+ | |||
+ | ~STEMbreezy | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/q4hJk1HYxDk | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/BzKZSwxJHJ0 | ||
+ | |||
+ | ~harungurcan | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2022|num-b=9|num-a=11}} | {{AMC8 box|year=2022|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 04:59, 22 January 2024
Contents
Problem
One sunny day, Ling decided to take a hike in the mountains. She left her house at , drove at a constant speed of
miles per hour, and arrived at the hiking trail at
. After hiking for
hours, Ling drove home at a constant speed of
miles per hour. Which of the following graphs best illustrates the distance between Ling’s car and her house over the course of her trip?
Solution 1 (Analysis)
Note that:
- At
Ling's car was
miles from her house.
- From
to
Ling drove to the hiking trail at a constant speed of
miles per hour.
It follows that at
Ling's car was
miles from her house.
- From
to
Ling did not move her car.
It follows that at
Ling's car was still
miles from her house.
- From
Ling drove home at a constant speed of
miles per hour. So, she arrived home
hour later.
It follows that at
Ling's car was
miles from her house.
Therefore, the answer is
~MRENTHUSIASM
Solution 2 (Elimination)
Ling's trip took hours, thus she traveled for
miles. Choices
,
, and
are eliminated.
Ling drove
miles per hour (mph) to the mountains, and
mph back to her house, so the rightmost slope must be steeper than the leftmost one. Choice
is eliminated.
This leaves us with
.
Solution 3 (Elimination)
Using the formula, and plugging in the values
mph and
hrs, we get that the distance from Ling's house to the mountains is
miles. That means the first slope ends at
so choices
and
are eliminated.
Furthermore, if the distance is miles, and Ling is returning home at
mph, it must have taken her
hours. Adding
and
(how long she hiked for) to her arrival time,
AM, we see she must have come back home at
PM. Choice
is eliminated, so the only valid choice left is choice
~ProProtractor
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/oUEa7AjMF2A?si=GriQR9m9WenwQVdf&t=1311
~Math-X
Video Solution (CRITICAL THINKING!!!)
~Education, the Study of Everything
Video Solution
https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=733
~Interstigation
Video Solution
https://youtu.be/1xspUFoKDnU?t=332
~STEMbreezy
Video Solution
~savannahsolver
Video Solution
~harungurcan
See Also
2022 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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.