SOLUTION: A cargo service operates by running a ship between Port A and Port B, a distance of 70km, at a constant speed of V kilometres per hour. For a given V the cost per hour of running t

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Question 1174143: A cargo service operates by running a ship between Port A and Port B, a distance of 70km, at a constant speed of V kilometres per hour. For a given V the cost per hour of running the ship is 9000 + 10V^2 dollars. Find the value of V which minimises the cost of the trip.
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!



The thing to notice here is that as V increases, the time t for the trip from port A to port B decreases.  



The total cost of the journey is the operating cost per hour times the number of hours:  C = +%289000%2B10V%5E2%29%2At+



The time for one journey is  t = 70km/V



Substitute this into the cost eqn:



C = +%289000%2B10V%5E2%29%2A%2870%2FV%29+

C = +%28630000%2FV+%2B+700%2AV%29+ 



Differentiate wrt V:

dC/dV = +-630000%2FV%5E2+%2B+700+ 



Set this to zero to find critical point (minimum)

+-630000%2FV%5E2+%2B+700+ = +0+

++V%5E2+=+900+   



+highlight%28V+=+30%29+ km/hr   (only kept the positive root)  





Spot Check:



V          t = 70/V         C = (9000+10V^2)*t

--         --------         ------------------

30km/hr    7/3 hr              $42000

29km/hr    70/29 hr            $42024

31km/hr    70/31 hr            $42023