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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]]}}


== 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, using the given length of the trail, <math>4.2</math>, and the given vertical increase, <math>4000</math> , we get a final answer of <math>\boxed{\textbf{(B) }246}.</math>
Solution by [[User:Juwushu|juwushu]].
Minor edits by ParticlePhysics and TigerSenju
==Solution 2==
Alternatively, observe that using <math>a=10x</math> and <math>b=\frac{y}{100}</math> makes the numbers much 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>.
~
==Video Solution==
https://youtu.be/l3VrUsZkv8I
~MC
==Video Solution by Central Valley Math Circle==
https://youtu.be/UewevLQ9qqE
~mr_mathman
== 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 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 TheBeautyofMath==
For AMC 10: https://youtu.be/uKXSZyrIOeU?t=540
For AMC 12: https://youtu.be/zaswZfIEibA?t=540
~IceMatrix
==Video Solution by Dr. David==
https://youtu.be/mrlTB_0QNyI
==Video Solution by yjtest==
https://www.youtube.com/watch?v=kyDHwV_KEy4
==Video solution by TheNeuralMathAcademy==
https://www.youtube.com/watch?v=4b_YLnyegtw&t=158s
==See Also==
{{AMC10 box|year=2024|ab=A|before=[[2023 AMC 10B Problems]]|after=[[2024 AMC 10B Problems]]}}
* [[AMC 10]]
* [[AMC 10 Problems and Solutions]]
* [[Mathematics competitions]]
* [[Mathematics competition resources]]
{{MAA Notice}}

Latest revision as of 18:11, 3 November 2025

The following problem is from both the 2024 AMC 10A #2 and 2024 AMC 12A #2, so both problems redirect to this page.

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 $T=aL+bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimates it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet?

$\textbf{(A) }240\qquad\textbf{(B) }246\qquad\textbf{(C) }252\qquad\textbf{(D) }258\qquad\textbf{(E) }264$

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 $a$ and $b$ gives you that $a=30$ and $b=\frac{3}{100}$. These values can be plugged back in showing that these values are correct. Now, using the given length of the trail, $4.2$, and the given vertical increase, $4000$ , we get a final answer of $\boxed{\textbf{(B) }246}.$

Solution by juwushu.

Minor edits by ParticlePhysics and TigerSenju

Solution 2

Alternatively, observe that using $a=10x$ and $b=\frac{y}{100}$ makes the numbers much 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 $15+8=12+11=23$, and since $23\cdot 3=69$, we determine that $x=y=3$.

The question asks us for $4.2a+4000b=42x+40y$. Since $x=y$, we have $(40+42)\cdot 3=\boxed{\textbf{(B) }246}$.

~

Video Solution

https://youtu.be/l3VrUsZkv8I ~MC

Video Solution by Central Valley Math Circle

https://youtu.be/UewevLQ9qqE

~mr_mathman

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

~Education, the Study of Everything

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 TheBeautyofMath

For AMC 10: https://youtu.be/uKXSZyrIOeU?t=540

For AMC 12: https://youtu.be/zaswZfIEibA?t=540

~IceMatrix

Video Solution by Dr. David

https://youtu.be/mrlTB_0QNyI

Video Solution by yjtest

https://www.youtube.com/watch?v=kyDHwV_KEy4

Video solution by TheNeuralMathAcademy

https://www.youtube.com/watch?v=4b_YLnyegtw&t=158s

See Also

2024 AMC 10A (Problems • Answer Key • Resources)
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2023 AMC 10B Problems 		Followed by
2024 AMC 10B Problems
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All AMC 10 Problems and Solutions

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