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Thread: a nice question about the speed of a ship

  1. #1
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    a nice question about the speed of a ship

    Hi folks,

    A ship is to make a voyage of 200 km at a constant speed. When the speed of the ship is v km/h the cost is $(v^2 + \dfrac{4000}{v}) $ USD per hour.
    Find the speed at which the ship should travel so that the cost of the voyage is a minimum.

    So, v = 200/t where t is the time in hours.

    Cost

    $C =v^2 + \dfrac{4000}{v}$

    $C = (\dfrac{200}{t})^2 + \dfrac{4000t}{200}$

    $C = \dfrac{40000}{t^2} + 20t$

    $\dfrac{dC}{dt} = \dfrac{-80000}{t^3} + 20 = 0$

    $t^3 = 4000$

    $t = 15.87$ hours

    $V = 200/15.987 = 12.6$ km/h

    this is the wrong answer. My book has 20km/h.
    Last edited by s_ingram; Jun 4th 2018 at 12:00 AM.
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  2. #2
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    Re: a nice question about the speed of a ship

    the total cost of the voyage is

    $\displaystyle \left(v^2+\frac{4000}{v}\right)\frac{200}{v} $
    Thanks from SlipEternal and Vinod
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  3. #3
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    Re: a nice question about the speed of a ship

    I see! My method doesn't even make sense. The rate of change of cost, has nothing to do with minimising the cost of voyage.
    Many thanks.
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