AMC 8 2006 Problem 13

AMC8 2006 年第 13 题

行程问题★★★☆☆

Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet?

Cassie 早上 8：30 离开 Escanaba，骑自行车前往 Marquette.她以每小时 12 英里的均匀速度骑自行车。Brian 早上 9 点离开 Marquette，骑着自行车前往 Escanaba。他以每小时 16 英里的均匀速度骑自行车。他们俩骑自行车使用的都是在 Escanaba 和 Marquette 之间的同一条 62 英里长的路线。则他们早上什么时候相遇？

A.10: 00
B.10: 15
C.10: 30
D.11: 00
E.11: 30

正确答案 · Correct Answer

解析 · Solution

Solution 1

If Cassie leaves

\displaystyle \frac{1}{2}

an hour earlier then Brian, when Brian starts, the distance between them will be

\displaystyle 62- \frac{12}{2} = 56

. Every hour, they will get 12+16=28 miles closer.

\displaystyle \frac{56}{28} = 2

, so 2 hours from

9:00

AM is when they meet, which is

11: 00

. Let

x

be the # of hours after Cassie leaves, when both of them meet. In that

x

hours, Cassie will travel

12x

miles. But, Brian will travel

\displaystyle 16(x-\frac{1}{2})

miles as he starts 1/2 hours after Cassie. These two distances sum to the total distance of 62 miles as they meet here, yielding the equation,

\displaystyle 12x+16(x-\frac{1}{2})=62

. After distribution 16 we get 12x+16x-8=62. After combining like terms and adding 8 to both sides we get 28x=70, and x=5/2. So they meet 5/2 hours after 8:30 a.m. which is

11: 00

. - LearnForEver

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