AMC 8 2025 Problem 19
AMC8 2025 年第 19 题
行程问题★★★☆☆
Two towns, and , are connected by a straight road that is 15 miles long. Travelling from city to town , the speed limit changes every 5 miles: from 25 to 40 to 20 miles per hour (mph). Two cars, one at town and one at town , start moving toward each other at the same time. They drive at exactly the speed limit in each portion of the road. How far from town , in miles, will the two cars meet?
A镇和B镇之间由一条15英里长的笔直公路相连。从A镇行驶到B镇，限速每5英里变化一次：依次为25英里/小时、40英里/小时和20英里/小时。两辆汽车，一辆在A镇，另一辆在B镇，它们同时出发相向而行。两车在各路段上都严格按照限速行驶。请问两车将在距离A镇多少英里处相遇？
- A.7.75
- B.8
- C.8.25
- D.8.5
- E.8.75
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
The first car, moving from town at 25 miles per hour, takes minutes. The second car, traveling another 5 miles from town , takes minutes. The first car has traveled for 3 minutes or th of an hour at 40 miles per hour when the second car has traveled 5 miles. The first car has traveled miles from the previous 5 miles it traveled at 25 miles per hour. They have 3 miles left, and they travel at the same speed, so they meet 1.5 miles through, so they are 5 + 2 + 1.5 = 8.5 miles from town .
From the answer choices, the cars will meet somewhere along the 40 mph stretch. Car travels 25 mph for 5 miles, so we can use dimensional analysis to see that it will be of an hour for this portion. Similarly, car spends of an hour on the 20 mph portion.
Suppose that car travels miles along the 40 mph portion-- then car travels 5-x miles along the 40 mph portion. By identical methods, car travels for hours, and car travels for hours.
At their meeting point, cars and will have traveled for the same amount of time, so we have $$$$$$$$ so 2x = 7, and x = 3.5 miles. This means that car will have traveled 5 + 3.5 = 8.5 miles.
-Benedict T (countmath1)
Instead of using algebra, we can visualize how far each car has traveled every hour. Let us divide each portion of the road into distances traveled each hour: For the 25 mph portion, we divide it up into 5 sections, because . Similarly, we divide the 40 mph portion into 8 sections ( ) and the 20 mph portion into 4 sections ( ). Thus, we have the following diagram.
After four hours, car A has reached the end of the 25 mph portion, while car B has traveled of the 20 mph portion. We can plot the amount of distance traveled, with the red dot representing car A and the blue dot representing car B.
If we keep moving the dots, they will eventually meet at segment 3.5 of the 40 mph portion of the road (note that each segment represents 1 hour of time traveled):
Now, we must account for the 5 miles in the 25 mph portion. Since the two cars meet at segment 3.5 of the 40 mph portion, we add the 5 miles traveled in the 25 mph section:
miles.
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