Question

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

$4$

$5$

$6$

$7$

$8$

Solution:

Number the pails so Michael is at pail $0$ and the truck at pail $1.$ Michael reaches pail $n$ at $40n$ seconds. The truck leaves pail $n$ at $50(n-1)$ seconds and arrives there at $50(n-1)-30$ seconds.

Michael and the truck are together at pail $n$ when $50(n-1)-30\le 40n\le 50(n-1),$ which simplifies to $5\le n\le 8.$

At pail $5$ they meet as the truck departs, at pails $6$ and $7$ Michael passes it, and at pail $8$ they meet as the truck arrives. Between pails $6$ and $7$ the truck must overtake Michael once more, so in total they meet $5$ times.

Thus, the correct answer is B.

2008 AMC 10B Problem 25

Solution:

Problem 25 in Other Years