AMC 8 2019 Problem 16
AMC8 2019 年第 16 题
行程问题★★★☆☆
Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?
Qiang 以 30 英里每小时的平均速度行驶了 15 英里。问他还需要以 55 英里每小时的速度行驶多少英里，这样可以使得全程的平均速度为 50 英里每小时？
- A.45
- B.62
- C.90
- D.110
- E.135
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
To calculate the average speed, simply evaluate the total distance over the total time. Let the number of additional miles he has to drive be $x.$ Therefore, the total distance is 15+x and the total time (in hours) is We can set up the following equation: Simplifying the equation, we get Solving the equation yields x=110, so our answer is 110.
If he travels 15 miles at a speed of 30 miles per hour, he travels for 30 min. Average rate is total distance over total time so (15+d)/(0.5 + t) = 50, where d is the distance left to travel and t is the time to travel that distance. Solve for to get d = 10+50t. You also know that he has to travel 55 miles per hour for some time, so d=55t. Plug that in for d to get 55t = 10+50t and t=2 and since d=55t, , the answer is 110.
Let be the amount of hours Qiang drives after his first 15 miles. Average speed, which we know is 50 mph, means total distance over total time. For 15 miles at 30 mph, the time taken is hour, so the total time for this trip would be hours. For the total distance, 15 miles are traveled in the first part and miles in the second. This gives the following equation:
Cross multiplying, we get that 15 + 55h = 50h + 25, and simple algebra gives h=2. In 2 hours traveling at 55 mph, the distance traveled is , which is choice 110.
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