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A town A is located on a river. We have to send cargo to town B which is located 'a' kilometers downstream and 'd' kilometers from the river. Government

wants to construct a sea link between B and the river such that the cost of transportation of goods from A to B is the cheapest. The transport cost of a unit of cargo per kilometer by waterway is half the cost incurred by taking the highway.

Your task is to help the government find a point in the river from which to construct a highway to town B so that the government's objective of reducing transportation cost is achieved. More specifically, calculate the distance from town A where the highway has to be constructed and the length of the highway to be constructed.



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    Let the transport take place from A along the shore of the river for  (a - x) km.  Then the distance along the highway on road will be  √(x² + d²).  let the cost per km of transport be  p for the waterway.

 So the total cost function is :  C(x) =  (a - x) p + 2p * √(x²+d²) 
         derivative wrt x:    C'(x) = - p + 2 p x /√(x²+d²)
                   C'(x) = 0  for  4 x² = x² + d²
                     =>  x = d/√3
     find  C"(x) = 2 p /√(x²+d²)  - p x* 2x / (x²+d²)³/²
                   = [2 p (x² +d²) - 2 p x² ]/ (x²+d²)³/²
   since C" is positive, the value of x gives the minimum value of cost function.

          length of the high way :  = √(x²+d²) = 2 d/√3
Distance from town A, along the river , from where the highway is constructed:
         = a - x = a - d/√3

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