SOLUTION: Calculus Optimization Question An oil refinery is located on the north bank of a river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks loc

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Question 1142710: Calculus Optimization Question
An oil refinery is located on the north bank of a river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying the pipe is $400,000/km over land to a point P on the north bank and $800,000/km under the river to the tanks. To minimize the cost of the pipeline, where should P be located?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let x be the distance from the refinery to the point P on the north bank of the river. Then the distance from point P to the storage tanks is the hypotenuse of a right triangle with legs 2 and sqrt(6-x). Then the cost function is

C+=+400000x+%2B+800000%2Asqrt%28%286-x%29%5E2%2B2%5E2%29
C+=+400000x+%2B+800000%2A%28x%5E2-12x%2B40%29%5E%281%2F2%29

Find the minimum cost by finding where the derivative is zero.

dC%2Fdx+=+400000+%2B+800000%28%281%2F2%29%28x%5E2-12x%2B40%29%5E%28-1%2F2%29%282x-12%29%29
dC%2Fdx+=+400000+%2B+800000%28%28x-6%29%2Fsqrt%28%28x%5E2-12x%2B40%29%29%29

Setting the derivative equal to 0 and solving...

400000+%2B+800000%28%28x-6%29%2Fsqrt%28%28x%5E2-12x%2B40%29%29%29+=+0
%28x-6%29%2Fsqrt%28%28x%5E2-12x%2B40%29%29+=+-1%2F2
%28x%5E2-12x%2B36%29%2F%28x%5E2-12x%2B40%29+=+1%2F4
4x%5E2-48x%2B144%29+=+x%5E2-12x%2B40
3x%5E2-36x%2B104+=+0


To four decimal places that is 4.8453.

ANSWER: Point P should be located about 4.8453km east of the refinery to minimize the cost of the pipeline.