2008 AMC 10B Problems/Problem 25

Problem

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 for the next pail. How many times will Michael and the truck meet?

$\mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 8$

Solution

The truck always moves for $20$ seconds, then stands still for $30$ . In these $50$ seconds, the truck will drive $200+0=200$ meters. In those $50$ seconds Michael will walk $250$ meters. So ultimately Michael will be way too far ahead of the truck for any more meetings to happen.

The movement of Michael and the truck is plotted below: Michael in blue, the truck in red. We can easily verify that indeed there will be $\boxed{5}$ meetings:

Michael will catch and overtake the truck while it is standing at the first pail.
The truck will start moving again and on its way to the second pail it will overtake Michael.
While the truck is standing at the second pail, Michael will walk past it.
The last meeting will occur exactly when both Michael and the truck arrive at the same time to the third pail.

$[asy] import graph; size(400,300,IgnoreAspect); real[] xt={0,20,50,70,100,120,150,170,200}; real[] yt={0,200,200,400,400,600,600,800,800}; real[] xm={0,200}; real[] ym={0,1000}; draw(graph(xt,yt),red); draw(graph(xm,ym),blue); xaxis("$time$",Bottom,LeftTicks); yaxis("$position$",Left,LeftTicks); [/asy]$

2008 AMC 10B (Problems • Answer Key • Resources)
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2008 AMC 10B Problems/Problem 25

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