1989 AHSME Problems/Problem 12

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Problem

The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. Which of the following is closest to the number of westbound vehicles present in a 100-mile section of highway?

$\textrm{(A)}\ 100\qquad\textrm{(B)}\ 120\qquad\textrm{(C)}\ 200\qquad\textrm{(D)}\ 240\qquad\textrm{(E)}\ 400$

Solution

At the beginning of the five-minute interval, say the eastbound driver is at the point $x=0$, and at the end of the interval at $x=5$, having travelled five miles. Because both lanes are travelling at the same speed, the last westbound car to be passed by the eastbound driver was just west of the position $x=10$ at the start of the five minutes. The first westbound car to be passed was just east of $x=0$ at that time. Therefore, the eastbound driver passed all of the cars initially in the stretch of road between $x=0$ and $x=10$. That makes $20$ cars in ten miles, so we estimate $200$ cars in a hundred miles.

$[asy] dot((0,0));dot((25,0));dot((50,0)); draw((15,5)--(5,5),EndArrow); draw((45,5)--(35,5),EndArrow); draw((0,0)--(50,0),dashed); draw((0,-5)--(10,-5),EndArrow); label("0",(0,0),W); label("10",(50,0),E); label("5",(25,0),S); [/asy]$