Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.5 AIME Find the answer United States

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?

Pick one

Solution

The given data from the first two hikes yield the system of equations
$\begin{aligned} 1.5a + 800b &= 69 \\ 1.2a + 1100b &= 69. \end{aligned}$
To solve this system, first subtract the second equation from the first equation to get $0.3a - 300b = 0$ , which implies $a = 1000b$ . Then the first equation becomes $1500b + 800b = 69$ , from which $b = \frac{69}{2300} = 0.03$ , and $a = 30$ . Therefore the model is $T = 30L + 0.03G$ . Substituting the values for the third hike gives $T = 30 \cdot 4.2 + 0.03 \cdot 4000 = 246$ minutes.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.