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Calculus

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An offshore well is 2 miles off the coast. The refinery is 4 miles down the coast. If a laying pipe in the ocean is twice as expensive as on land, what path the pipe should follow in order to minimize the coast.

Is the answer 2.1? Can you some list their steps?

  • Calculus -

    Make a sketch, mark the well as W and the point on the shore closest to W as A, label the refinery as R
    Place P somewhere between A and R.
    They will lay the pipeline from R to P along the shore, then directly through the water to W
    let AP=x , the PR = 4-x, and AW=4
    by Pythagoras:
    WP = √(9-x^2) or (9-x^2(^(1/2)

    cost = 1(4-x) + 2(9-x^2)^(1/2) , the actual money units don't matter as long as they are in the ratio 1 : 2

    d(cost)/dx = -1 + (1/2)(9 - x^2)^(-1/2) (-2x)
    = 0 for a min of cost
    1 = x/√(9-x^2)
    x = √(9-x^2)
    square both sides
    x^2 = 9-x^2
    2x^2 = 9
    x^2 = 4.5
    x = √4.5 = appr 2.12

    So they should aim for a point P which is 2.12 miles from A or 1.88 miles from R

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