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Calculus/Calculus- Optimization

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Expert: - 11/18/2008


Question
An oil refinery is located on the north bank of a straight river that is 1 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 7 km east of the refinery. The cost of laying pipe is $500,000 per km over land to a point P on the north bank and $1,000,000 per km under the river to the tanks. To minimize the cost of the pipeline, how far from the refinery should P be located?

Answer
Refinery: 1 km up river

Pipleline from refinery to other bank, 7km away.

Cost for land is $500K/km; cost in water is $1,000K.

Location of turn into the water: x meters up river from refinery.

Distance to storage tanks: 7-x.

Distance across river to refinery: √(x²+1²).

Total cost: (7-x)500 + √(x²+1²)1000.

Take derivative with respect to x of the total cost, set it equal to zero, and solve.

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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

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