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Math Help - Optimisation Problem - Ships

  1. #1
    Junior Member
    Joined
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    Optimisation Problem - Ships

    Hello everyone,

    I am having trouble with this problem, and I have made my problem and work available at ImageShack - Image Hosting :: shipoptimisationprobbx7.jpg (The diagram is my own, not given by the question)


    I was asking my friend about this problem and took a screenshot of my Windows screen.

    Thank you very much for the help!
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, scherz0!

    I had difficulty reading some of the data.
    I hope I got it right . . .


    Ship A looks northeast and sees ship B 5 km away.
    Ship A is moving east at 5 km/hr; ship B is moving south at 4 km/hr.
    When will they be at their closest?

    At the very beginning, the diagram looks like this:
    Code:
                          * Q
                        * |
                      *   |
                    *     |
               5  *       | 5√2/2
                *         |
              *           |
            * 45         |
        P * - - - - - - - * R
               5√2/2
    Ship A is at P; ship B is at Q.\quad PQ \:=\: 5,\;PR \:=\: \tfrac{5\sqrt{2}}{2},\;QR \:=\: \tfrac{5\sqrt{2}}{2}




    Code:
                          * Q
                        * |
                      *   | 4t
                    *     |
               5  *       * B
                *       * |
              *       *   | 5√2/2-4t
            *       *     |
        P * - - - * - - - * R
             5t   A 5√2/2-5t
    In t hours, ship A has moved 5t km to point A.\quad AR \:=\:\tfrac{5\sqrt{2}}{2} - 5t

    In the same t hours, ship B has move 4t km to point B.\quad BR \:=\:\tfrac{5\sqrt{2}}{2} - 4t


    Their distance is given by: . D \;=\;AB^2 \;=\;\left(\tfrac{5\sqrt{2}}{2} - 5t\right)^2 + \left(\tfrac{5\sqrt{2}}{2} - 4t\right)^2

    . . which simplifies to: . D \;=\;41t^2 - 45\sqrt{2}t + 25


    Differentiate and equate to zero: . 82t - 45\sqrt{2} \:=\:0 \quad\Rightarrow\quad t \:=\:\frac{45\sqrt{2}}{82}


    Therefore, they are closest at: . 0.776092809\text{ hours} \;\approx\;46\text{ mimutes, }34\text{ seconds}

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