2008 AMC 10B Problems/Problem 25

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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: (the graph of the truck should start 200 units higher)

  • 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]


See also

2008 AMC 10B (ProblemsAnswer KeyResources)
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