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Difference between revisions of "2024 AMC 10A Problems/Problem 2"
(→‎Solution 1)
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== Solution 1 ==
 
== Solution 1 ==
 
Plug in the values into the equation to give you the following two equations:
 
Plug in the values into the equation to give you the following two equations:
−
<cmath>69=1.5a+800b</cmath> <cmath>69=1.2a+1100b</cmath>
+
\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.
 
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>
 
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 by [[User:Juwushu|juwushu]].
 +
 
==See also==
 
==See also==
 
{{AMC10 box|year=2024|ab=A|num-b=1|num-a=3}}
 
{{AMC10 box|year=2024|ab=A|num-b=1|num-a=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 16:09, 8 November 2024

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, use the given $4.2$ mile length and $4000$ foot change in elevation, giving you a final answer of $\boxed{\textbf{(B) }246}.$

Solution by juwushu.

See also

2024 AMC 10A (Problems • Answer Key • Resources)
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Problem 1 		Followed by
Problem 3
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All AMC 10 Problems and Solutions

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