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Optimization of fuel consumption question

  1. Sep 15, 2011 #1
    1. The problem statement, all variables and given/known data

    The fuel consumption of a river boat is kv3 litres per hour where k is a constant and v km/h is it's speed through the water throughout this question v > 4 km/h.

    i) determine the fuel consumption for a trip of x km against a current of 4km/h and find the speed at which the fuel consumption is minimised.

    ii) determine the most economical speed for a trip of x km against the current and return, assuming that the current is constant at 4km/h and that the same speed through the water is maintained in each direction.

    2. Relevant equations

    none in particular

    3. The attempt at a solution

    for part (i) is simply got

    kv3 = k(4 + (x/t))3, where is x is the distance travelled against the current and t is the time taken to travel it. However every time I try to optimize it I get (x/t) = 0 , which means that the boat does not move a distance x at all! Is this an acceptable answer then? To say that fuel consumption is optimised when the boat does not move against the current at all?

    I have not attempted the second part yet...

    thanks
     
  2. jcsd
  3. Sep 15, 2011 #2

    dynamicsolo

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    Homework Helper

    Consider that the distance x is measured relative to the stationary riverbanks, but the boat moves at a different speed relative to them in each direction. So the total time is not simply x/v (it would be 2x/v by this reasoning anyway), but the sum of two times, one being x divided by the speed the boat makes relative to the riverbanks going against the current and the other time being x divided by the speed the boat makes going with the current. The fuel consumption would be kv3* times that total time and that is what need to be minimized.

    *v is the speed of the boat relative to the water, which we are to assume is unchanged in either direction of travel
     
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