2024 AMC 10A 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 $T=aL+bG,$ 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: $69=1.5a+800b$ $69=1.2a+1100b$ 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.

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2024 AMC 10A Problems/Problem 2

Problem

Solution 1