Question

一个用于估计沿小路登上山顶所需时间的模型形如 $T = aL + bG$ ，其中 $a$ 和 $b$ 为常数， $T$ 是分钟数， $L$ 是小路长度，单位为英里， $G$ 是海拔上升高度，单位为英尺。模型估计，若小路长 $1.5$ 英里并上升 $800$ 英尺，或小路长 $1.2$ 英里并上升 $1100$ 英尺，到达山顶都需要 $69$ 分钟。若小路长 $4.2$ 英里并上升 $4000$ 英尺，模型估计需要多少分钟？

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 estimate it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet?

$240$

$246$

$252$

$258$

$264$

答案：B

知识点：方程组一次方程

难度评级：990

解答：

两个方程为 $1.5a + 800b = 69$ 和 $1.2a + 1100b = 69$ 。相减消去 $69$ ，得到 $0.3a - 300b = 0$ ，所以 $a = 1000b$ 。代入第一式得 $1500b + 800b = 2300b = 69$ ，因此 $b = 0.03$ 、 $a = 30$ 。所求时间为 $T = 30(4.2) + 0.03(4000)$ $= 126 + 120$ $= 246$ 。所以正确答案是 B。

Subtract the two equations $1.5a + 800b = 69$ and $1.2a + 1100b = 69$ to kill the $69.$ That leaves $0.3a - 300b = 0,$ so $a = 1000b.$ Now substitute: $1500b + 800b = 2300b = 69,$ so $b = 0.03$ and $a = 30.$ Then $T = 30(4.2) + 0.03(4000)$ $= 126 + 120$ $= 246.$ Therefore, the answer is B.

2024 AMC 10A 第 2 题

答案：B

难度评级：990

解答：

其他年份的第 2 题