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

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) (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

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

Setting the derivative equal to 0 and solving...

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.

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