An Oil Rig is located a = 10 miles from a straight coastline. An Oil Refinery is

Question

An Oil Rig is located a = 10 miles from a straight coastline. An Oil Refinery is located 16 miles down the coastline. The cost of putting in a pipeline from the Oil Rig to the Oil Refinery depends on whether it is on land or in the water. Rather than coming directly to the shoreline and then turning at a Right Angle, suppose that the Oil Pipeline met the shore at a point b = 6 miles from the "Right Angle". If the cost is $10000 per mile for land and $17000 per mile for water, what is the cost of the Pipeline?

Explanation / Answer

Distance covered in land = 16 x

Distance covered in water = sqrt(100 + x^2)

So the equation becomes

cost, C = 10000(16 x) + 17000(sqrt(100 + x^2))

Differentiate wrt x & equate to zero

dC/dx = -10000 + 17000 * x (100 + x^2)^-1/2 = 0

17000* x (100 + x^2)^-1/2 = 10000

17* x = 10(100 + x^2)^1/2

Squaring both sides

289x^2 = 100(100 + x^2)

189x^2 = 10000

x = 7.274

Putting the value in C

C = 10000(16 - 7.274) + 17000(sqrt(100 + (7.274)^2))

C = 87260 + 210217.27

C = $ 297477.27

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