Optimization of fuel consumption question

[animboy]

Homework Statement

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.

Homework Equations

none in particular

The Attempt at a Solution

for part (i) is simply got

kv³ = k(4 + (x/t))³, where is x is the distance traveled 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

 

[dynamicsolo]

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