Question

The cost of fuel in running an engine is proportional to the square of the speed in
km/h and is A64 per hour when the speed is 16 km/h. Other costs amount to A100
per hour. Find the speed, which would minimize the cost.

Answer 1

 rate of cost of fuel FC =  price of fuel  P *  volume of fuel consumed V
    
rate of cost of fuel FC is proportional to the square of speed.    So  volume V of fuel consumed per hour is proportional to the square of speed v  in kmph.

  rate of fuel cost FC = $ 64 /hour        for Speed v = 16 kmph
                 FC = K v²            , where K is the constant of proportionality.
          => $ 64 = K * 16²    =>  K = 0.25 $-Hr/km²

  rate of other costs OC =  $ 100 / hour

 total costs rate = TC = FC + OC = 0.25 v²  + 100      $ / hour

 Let the engine in the vehicle drive the vehicle for a distance of  S km.  Then the time taken to cover the distance S is :  T = S / v.    The total distance S to be travelled is a constant.
 
  Total cost for  T hours in $
     C   =  T * (0.25 v² + 100) 
          =  S / v * (0.25 v² + 100)
          = 0.25 S  v  + 100 S / v
 
differentiate C wrt   v  and find the value of v for which  C is minimum.

  d C / d v =  0.25 S - 100 S / v²
                     = 0    for minimum
           so  0.25 = 100 / v²
                 v² =  400
                   v = 20 kmph
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So the minimum cost of running the engine is :   0.25 *  20² + 100 = 200 $ / hour