2008 AMC 12B Problem 20

Attempt Problem 20 of the 2008 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 12B solutions, or check the answer key.

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20.

Michael walks at the rate of 55 feet per second on a long straight path. Trash pails are located every 200200 feet along the path. A garbage truck travels at 1010 feet per second in the same direction as Michael and stops for 3030 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?

44

55

66

77

88

Answer: B
Concepts:relative speeddistance rate and timeprocess simulation
Difficulty rating: 1860
Solution:

Number the pails so Michael is at pail 00 and the truck at pail 11 at time 0.0. Michael reaches pail nn at 40n40n seconds. The truck spends 2020 seconds between pails and 3030 stopped, so it leaves pail nn at 50(n1)50(n - 1) seconds and (for n2n \ge 2) arrives at 50(n1)30.50(n - 1) - 30.

Michael is at pail nn while the truck is there exactly when 50(n1)3040n50(n-1) - 30 \le 40n 50(n1),\le 50(n-1), which simplifies to 5n8.5 \le n \le 8. So they meet at pail 55 (at t=200,t = 200, as the truck departs), pail 66 (t=240t = 240), pail 77 (t=280t = 280), and pail 88 (t=320,t = 320, as the truck arrives).

Between pails 66 and 77 the truck (moving at 1010 ft/s) pulls ahead of and is then overtaken by Michael once more, adding one crossing. In all, they meet 55 times.

Thus, the correct answer is B.

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