A pipeline is to be constructed to connect a point P on the shore of one side of

Question

A pipeline is to be constructed to connect a point P on the shore of one side of a straight river to a point Q which is located on the shore on the other side of the river and 127 metres down the river. The cost per meter of constructing the pipeline over was is 3 times the cost of constructing the pipeline over land. The plan is to run the pipeline from point P along the shore x metres and then cross the river at an angle so as to connect directly to point Q. The river is 49 metres wide. For what choice of distance x is the cost minimum.

Explanation / Answer

Q | |- |-- |---- |----------- R ...B ...P QR = w = 49 m RB = x [ pl. note change of symbol to match with that of link ] BP = y = 127-x angle RQB = z ratio of cost water:land = k = 3 it can be shown that the optimal cost will be when sin z = 1/k (see link for derivation) which yields x = w/sqrt(k^2-1) = 49/sqrt(9-1) = 17.32 m y = 127 - x = 109.68 m =================

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