Question

艾米丽沿一条笔直长路骑自行车时，看到埃默森在她前方 $\frac{1}{2}$ 英里处同方向滑行。她超过他后，可以从后视镜里看到他，直到他落在她后方 $\frac{1}{2}$ 英里处。艾米丽以每小时 $12$ 英里的恒定速度骑行，埃默森以每小时 $8$ 英里的恒定速度滑行。艾米丽能看到埃默森多少分钟？

As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction $\frac{1}{2}$ mile in front of her. After she passes him, she can see him in her rear view mirror until he is $\frac{1}{2}$ mile behind her. Emily rides at a constant rate of $12$ miles per hour, and Emerson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Emerson?

$\ 6$

$\ 8$

$\ 12$

$\ 15$

$\ 16$

答案：D

知识点：相对速度路程、速度与时间单位换算

难度评级：1100

小提示：

在可见的时间段内，艾米丽相对埃默森总共多前进 $1$ 英里

During the visible interval Emily gains a total of $1$ mile on Emerson.

大提示：

使用艾米丽相对于埃默森的速度

Use Emily’s speed relative to Emerson’s speed.

解答：

设 $d$ 为艾米丽领先埃默森的距离。艾米丽能看到埃默森当且仅当 $-\dfrac 12 \leq d \leq \frac 12$ 。其中 $t$ 以小时为单位；令 $t= 0$ 时 $d = -\frac 12$ 。根据两人的相对速度， $d = (12-8)t - \frac 12$ 。当 $d = \frac 12$ 时， $\frac 12 = 4t - \frac 12 \implies 1 = 4t$ $\implies t = 0.25\text{。}$ 因此经过 $0.25$ 小时，也就是 $60\cdot0.25 = 15$ 分钟。

所以正确答案是 D。

Let $d$ be how far Emily is ahead of Emerson. Emily sees Emerson if $-\dfrac 12 \leq d \leq \frac 12.$ Suppose at $t= 0,$ where $t$ is in hours, that $d = -\frac 12.$ Then, $d = (12-8)t - \frac 12 .$ Since we must find where $d = \frac 12,$ we find the time where $\frac 12 = 4t - \frac 12 \implies 1 = 4t$ $\implies t = 0.25.$ Since $0.25$ hours passed, we know that $60\cdot0.25 = 15$ minutes passed.

Therefore, the answer is D .

2010 AMC 8 第 8 题

答案：D

难度评级：1100

小提示：

大提示：

解答：

其他年份的第 8 题