Question

Michael 在一条长直路上以每秒 $5$ 英尺的速度步行。路上每隔 $200$ 英尺有一个垃圾桶。一辆垃圾车以每秒 $10$ 英尺的速度沿同一方向行驶，并在每个垃圾桶处停 $30$ 秒。当 Michael 经过一个垃圾桶时，他注意到前方的垃圾车刚离开下一个垃圾桶。Michael 和垃圾车会相遇多少次？

Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck travels at $10$ feet per second in the same direction as Michael and stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?

$4$

$5$

$6$

$7$

$8$

答案：B

知识点：路程、速度与时间不等式过程模拟

难度评级：2090

解答：

给垃圾桶编号，使 Michael 在 $0$ 号桶，垃圾车在 $1$ 号桶。Michael 在 $n$ 号桶的到达时间为 $40n$ 秒；垃圾车在 $n$ 号桶的离开时间为 $50(n-1)$ 秒，到达时间为 $50(n-1)-30$ 秒。

Michael 和垃圾车在 $n$ 号桶相遇当且仅当 $50(n-1)-30\le 40n$ $\le 50(n-1)$ ，化简得 $5\le n\le 8$ 。

在 $5$ 号桶他们在垃圾车离开时相遇，在 $6$ 和 $7$ 号桶 Michael 经过停着的车，在 $8$ 号桶他们在垃圾车到达时相遇。此外，在 $6$ 号和 $7$ 号桶之间，垃圾车还必须超过 Michael 一次，所以总共相遇 $5$ 次。

所以正确答案是 B。

Number the pails so Michael is at pail $0$ and the truck at pail $1.$ Michael reaches pail $n$ at $40n$ seconds. The truck leaves pail $n$ at $50(n-1)$ seconds and arrives there at $50(n-1)-30$ seconds.

Michael and the truck are together at pail $n$ when $50(n-1)-30\le 40n$ $\le 50(n-1),$ which simplifies to $5\le n\le 8.$

At pail $5$ they meet as the truck departs, at pails $6$ and $7$ Michael passes it, and at pail $8$ they meet as the truck arrives. Between pails $6$ and $7$ the truck must overtake Michael once more, so in total they meet $5$ times.

Thus, the correct answer is B.

2008 AMC 10B 第 25 题

答案：B

难度评级：2090

解答：

其他年份的第 25 题