AMC12 2024 A

AMC12 2024 A · Q2

AMC12 2024 A · Q2. It mainly tests Systems of equations, Arithmetic misc.

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?

一个用于估计徒步爬到山顶所需时间的模型形式为 $T=aL+bG$，其中 $a$ 和 $b$ 是常数，$T$ 是分钟数，$L$ 是小路长度（英里），$G$ 是海拔上升（英尺）。该模型估计一条长 $1.5$ 英里、上升 $800$ 英尺的小路需要 $69$ 分钟爬到山顶；一条长 $1.2$ 英里、上升 $1100$ 英尺的小路也需要 $69$ 分钟。如果小路长 $4.2$ 英里、上升 $4000$ 英尺，该模型估计需要多少分钟爬到山顶？

(A) 240 240

(B) 246 246

(D) 258 258

(E) 264 264

Answer

Correct choice: (B)

正确答案：（B）

Solution

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, 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

将值代入方程，得到： $69=1.5a+800b$，$69=1.2a+1100b$。解得 $a=30$，$b=\frac{3}{100}$。验证这些值正确。现在，使用小路长度 $4.2$ 和垂直上升 $4000$，得到最终答案 $\boxed{\textbf{(B) }246}$。 Solution by juwushu。 Minor edits by ParticlePhysics and TigerSenju

Topics