An oil refinery is located 1 km north of the north bank of a straight river that

Question

An oil refinery is located 1 km north of the north bank of a straight 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 5 km east of the refinery. The cost of laying pipe is $200,000/km over land to a point P on the north bank and $400,000/km under the river to the tanks. To minimize the cost of the pipeline, how far downriver from the refinery should the point P be located? (Round your answer to two decimal places.) km Need Help? Read It Chat About It

Explanation / Answer

let p be the distance from refinery
The function defining the costs for the project is:
f(p)=200000*p+400000*sqrt(2^2+(5-p)^2)
where 200000*p is the costs for laying alongside the river to point p

and 400000*sqrt(2^2+(5-p)^2) is the costs for diagonally laying the pipe under the river from point P to the tanks (according to pythagoras)
differentiating f(p) we get

200000 + 400000(p-5)/(sqrt(4+(p-5)^2))=0

p=3.84 km.

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