Question

A model used to estimate the time it will take to hike to the top of a 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 estimate it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet?

$240$

$246$

$252$

$258$

$264$

Solution:

From $1.5a+800b=1.2a+1100b$ we get $0.3a=300b,$ so $a=1000b.$ Then $69=1.5a+800b=1500b+800b=2300b,$ giving $b=0.03$ and $a=30.$ For $L=4.2,\ G=4000,$ $T=30(4.2)+0.03(4000)=126+120=246.$ Thus, the correct answer is B.

2024 AMC 12A Problem 2

Solution:

Problem 2 in Other Years