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Math Help - Optimization; find my error

  1. #1
    Junior Member Tclack's Avatar
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    Optimization; find my error

    an offshore oil well located 5km from the shore. That point is 8km from the oil collection facility. It costs a million dollars per km to build piping in the water and 500,000 dollars per km to build piping on the shore. What location should you place the point where the shore and sea connection to minimize the cost of laying the piping?

    Cost = c

     c=1,000,000WP + 500,000PB

     WP=\sqrt{(WA)^2 + (AP)^2}

    AP= 8km-PB

    WP=\sqrt{25+(8-PB)^2}=\sqrt{89-16PB+(PB)^2}


     c=1,000,000\sqrt{89-16PB+(PB)^2} + 500,000PB

     dc/dPB = 1,000,000 \frac{-16+2PB}{2 \sqrt{89 - 16PB + (PB)^2}} + 500,000 = 0

     -\sqrt{89-16PB+(PB)^2}=-16+2PB

    89-16PB+(PB)^2 = 256-32PB+4(PB)^2

    3(PB)^2-16PB+167=0

    oops, The quadratic gives me a negative square root. Where did I go wrong?


    P.S. I've solved this done by turning AP into x and PB into 8-x. I get the solution that way. I was trying to do it as above and I cannot get it, Both methods should give me the answer, but THIS way won't work for some reason.
    Attached Thumbnails Attached Thumbnails Optimization; find my error-shore-facility.bmp  
    Last edited by mr fantastic; October 6th 2009 at 09:42 PM. Reason: Fixed a line of latex
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  2. #2
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    mr fantastic's Avatar
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    Quote Originally Posted by Tclack View Post
    an offshore oil well located 5km from the shore. That point is 8km from the oil collection facility. It costs a million dollars per km to build piping in the water and 500,000 dollars per km to build piping on the shore. What location should you place the point where the shore and sea connection to minimize the cost of laying the piping?

    Cost = c

     c=1,000,000WP + 500,000PB

     WP=\sqrt{(WA)^2 + (AP)^2}

    AP= 8km-PB

    WP=\sqrt{25+(8-PB)^2}=\sqrt{89-16PB+(PB)^2}


     c=1,000,000\sqrt{89-16PB+(PB)^2} + 500,000PB

     dc/dPB = 1,000,000\frac{-16+2PB}{2\sqrt{89-16PB+<br /> <br />
(PB)^2}}+500,000=0

     -\sqrt{89-16PB+(PB)^2}=-16+2PB

    89-16PB+(PB)^2 = 256-32PB+4(PB)^2

    3(PB)^2-16PB+167=0

    oops, The quadratic gives me a negative square root. Where did I go wrong?


    P.S. I've solved this done by turning AP into x and PB into 8-x. I get the solution that way. I was trying to do it as above and I cannot get it, Both methods should give me the answer, but THIS way won't work for some reason.
    For starters I'd make things easier by:

    1. Working in units of millions of dollars.

    2. Let PB = x.

    Then:

     c= WP + \frac{x}{2}

     WP=\sqrt{(WA)^2 + (AP)^2}

    AP= 8 - x

    WP =\sqrt{25+(8-x)^2} =\sqrt{89-16x + x^2}


     c= \sqrt{89-16x + x^2} + \frac{x}{2}

     \frac{dc}{dx} = \frac{-16+2x}{2\sqrt{89-16x + x^2}}+ \frac{1}{2}=0

     -\sqrt{89-16x+x^2}=-16+2x

    89-16x+x^2 = 256-{\color{red}64}x + 4x^2

    3x^2-{\color{red}48}x+167=0
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  3. #3
    Junior Member Tclack's Avatar
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    It's such a simple error. Thanks, but I think my setup may be wrong.
    This answer simplifies to  8 +/- \frac{5\sqrt{3}}{3}

    The actual answer is  +/-\frac{5}{\sqrt{3}}

    What's wrong with the setup?
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  4. #4
    Junior Member Tclack's Avatar
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    Ah, nevermind, This actually makes sense, If you multiply by

    \frac{\sqrt{3}}{\sqrt{3}}

    You get 8+/- \frac{5}{\sqrt{3}}

    Which makes complete sense. 8 being the km. The way my book solves it, uses line AP as x, and my answer is 8-x.
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