Math

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Power Plant on the River Problem

As the construction manager for the Lightem Up Power Company , you make decisions to hep minimize costs of the company's construction projects. Today you've been asked to determine the path of a pipe that will house communication lines from the factory to the power plant pictured here.

Factory . F_______7 miles_______
| river |
4 miles
|______________________|
.P Power
Plant

You must connect a pipe from point F at the factory to point P at the power plant. It will cost $12 per foot to lay the pipe on land and $21 per foot to lay the pipe in the water. Where should the pipe be placed so as to minimize the total cost of laying the pipe?

  • Math -

    let the point directly across from P (the power plant) be Q.
    then FQ = 7 miles

    let R be a point between F and Q so that RQ = x miles, and FR = 7-x miles.
    RP^2 = x^2 + 16
    RP = (x^2+16)^(1/2)

    So the path of the pipeline is PR + RF.
    Cost of pipe on land: $12 per foot = $12(5280) per mile
    = $63360 per mile
    Cost of pipe in water: $21 per foot or
    $110880 per mile

    Cost = 110880*RP + 63360*FR
    = 110880(x^2+16)^)1/2) + 63360(7-x)

    d(Cost)/dx = (1/2)(2x)(110880)(x^2+16)^(-1/2) - 63360x
    = 0 for a minimus of Cost
    this reduced to

    7x/√(x^2+16) = 4 (I divided by 15840x)
    cross-multiplying and squaring both sides gave me
    49x^2 = 16x^2 + 256
    33x^2 = 256
    x = 16/√33 = 2.785

    then 7-x = 4.215

    They should aim for a point 4.215 miles from the factory, then cross the river to the power plant.

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