Difference between revisions of "2024 AMC 10A Problems/Problem 2"

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{{duplicate|[[2024 AMC 10A Problems/Problem 2|2024 AMC 10A #2]] and [[2024 AMC 12A Problems/Problem 2|2024 AMC 12A #2]]}}
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#redirect [[2024 AMC 12A Problems/Problem 2]]
 
 
== Problem ==
 
 
 
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form <math>T=aL+bG,</math> where <math>a</math> and <math>b</math> are constants, <math>T</math> is the time in minutes, <math>L</math> is the length of the trail in miles, and <math>G</math> is the altitude gain in feet. The model estimates that it will take <math>69</math> minutes to hike to the top if a trail is <math>1.5</math> miles long and ascends <math>800</math> feet, as well as if a trail is <math>1.2</math> miles long and ascends <math>1100</math> feet. How many minutes does the model estimates it will take to hike to the top if the trail is <math>4.2</math> miles long and ascends <math>4000</math> feet?
 
 
 
<math>\textbf{(A) }240\qquad\textbf{(B) }246\qquad\textbf{(C) }252\qquad\textbf{(D) }258\qquad\textbf{(E) }264</math>
 
 
 
== Solution 1 ==
 
Plug in the values into the equation to give you the following two equations:
 
\begin{align*}
 
69&=1.5a+800b, \
 
69&=1.2a+1100b.
 
\end{align*}
 
Solving for the values <math>a</math> and <math>b</math> gives you that <math>a=30</math> and <math>b=\frac{3}{100}</math>. These values can be plugged back in showing that these values are correct.
 
Now, use the given <math>4.2</math>-mile length and <math>4000</math>-foot change in elevation, giving you a final answer of <math>\boxed{\textbf{(B) }246}.</math>
 
 
 
Solution by [[User:Juwushu|juwushu]].
 
 
 
==Solution 2==
 
Alternatively, observe that using <math>a=10x</math> and <math>b=\frac{y}{100}</math> makes the numbers much more closer to each other in terms of magnitude.
 
 
 
Plugging in the new variables:
 
\begin{align*}
 
69&=15x+8y, \
 
69&=12x+11y.
 
\end{align*}
 
 
 
The solution becomes more obvious in this way, with <math>15+8=12+11=23</math>, and since <math>23\cdot 3=69</math>, we determine that <math>x=y=3</math>.
 
 
 
The question asks us for <math>4.2a+4000b=42x+40y</math>. Since <math>x=y</math>, we have <math>(40+42)\cdot 3=\boxed{\textbf{(B) }246}</math>.
 
 
 
~Edited by Rosiefork
 
 
 
== Video Solution by Math from my desk ==
 
 
 
https://www.youtube.com/watch?v=ENbD-tbfbhU&t=2s
 
 
 
== Video Solution (🚀 2 min solve 🚀) ==
 
 
 
https://youtu.be/OmaG3iG7xFs
 
 
 
<i>~Education, the Study of Everything</i>
 
 
 
==Video Solution by Number Craft==
 
 
 
https://youtu.be/k1rTBtiDWqY
 
 
 
== Video Solution by Daily Dose of Math ==
 
 
 
https://youtu.be/W0NMzXaULx4
 
 
 
~Thesmartgreekmathdude
 
 
 
== Video Solution by Power Solve ==
 
https://youtu.be/j-37jvqzhrg?si=2zTY21MFpVd22dcR&t=100
 
 
 
==Video Solution by SpreadTheMathLove==
 
https://www.youtube.com/watch?v=6SQ74nt3ynw
 
 
 
==Video Solution by FrankTutor==
 
https://youtu.be/A72QJN_lVj8
 
 
 
==Video Solution by Serenaa==
 
https://www.youtube.com/watch?v=i10tEEHq0sI&t=183s
 
 
 
==See also==
 
{{AMC10 box|year=2024|ab=A|num-b=1|num-a=3}}
 
{{AMC12 box|year=2024|ab=A|num-b=1|num-a=3}}
 
{{MAA Notice}}
 

Revision as of 21:26, 20 March 2025