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| A Geometric Race (Posted on 2007-05-02) |
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Clement challenges Dalton to a five kilometer car race. Dalton drives at a constant speed of 22 kilometers per hour throughout the race.
Clement waits for precisely three seconds after Dalton has reached 600 meters from the start line, and then Clement starts his car, covering every xth kilometer at the rate of 2.5*3x kilometers per hour.
For example, Clement covers the first kilometer at the rate of 7.5 kilometers per hour , the second kilometer at the rate of 22.5 kilometers per hour and so on.
Who is the winner of the race?
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Submitted by K Sengupta
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Rating: 3.0000 (1 votes)
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Solution:
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(Hide)
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The remaining 5 – 0.6 = 4.4 kilometers is traversed by Dalton in 22/5*1/22*60 = 12 minutes, while the time taken by Clement to cover every xth kilometer is 60/(2.5*3^x) = 8/(3^(x - 1)) minutes where x is 1, 2, 3, 4, 5.
Accordingly the entire distance of five kilometers was covered by Clement in 8(1 + 1/3 + 1/3^2 + 1/3^3 + 1/3^4) = 11.95061728 minutes.
Since Clement has started his car precisely three seconds or 0.05 minutes after Dalton B has a start of 0.6 kilometers, it follows that the amount of time taken by Clement to traverse the entire distance = 11.95061728 + 0.05 = 12.00061728 minutes.
Hence, Clement reaches the destination precisely 0.00061728*60 = 0.0370368 seconds after Dalton and thus Dalton is the winner of the race.
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