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minimum value calculations
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An oil production platform is 9 sqrt3km directly offshore from the nearest point A on a straight coast. It is to be connected by a pipeline to an onshore refinery R at a distance 100km along the straight coastline from A. It costs �1 million per km to lay the pipeline onshore, and �2million per km oto lay it underwater.
By introducing one sensible unknown write an expression for the cost of the pipeline. Find the minimum value of this cost.
(if x is the unknown and y is the cost, find dy/dx=0. Explain why this is a minimum value.)
By introducing one sensible unknown write an expression for the cost of the pipeline. Find the minimum value of this cost.
(if x is the unknown and y is the cost, find dy/dx=0. Explain why this is a minimum value.)
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For more on marking an answer as the "Best Answer", please visit our FAQ.Let x = distance along the coast where pipe joins "land"
Taking costs in millions = y
Then y = (100 - x) + 2 sqrt(243 + x^2)
So dy/dx = -1 +2x sqrt (243 + x^2) ^ (-1)
Solving dy/dx = 0 for turning values gives x = +/-9
Discard the negative solution and take x = 9 km
So y(min) = 91 + 2sqrt (243 + 81) = 91 + 36 = 127 millions
Check that d2y/dx2 is positive at x = 9 for a local minimum
Taking costs in millions = y
Then y = (100 - x) + 2 sqrt(243 + x^2)
So dy/dx = -1 +2x sqrt (243 + x^2) ^ (-1)
Solving dy/dx = 0 for turning values gives x = +/-9
Discard the negative solution and take x = 9 km
So y(min) = 91 + 2sqrt (243 + 81) = 91 + 36 = 127 millions
Check that d2y/dx2 is positive at x = 9 for a local minimum
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