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Math Help - Find shortest time to travel

  1. #1
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    Find shortest time to travel

    Firstly, apologies for the hard-to-see image. It's just for an outline of the problem.

    The questions asks to find the value of x (on the right vertical side) to find the minimum time to travel to the top right corner of the "forest"

    The bush can be travelled through at 3km/h and the "clear" area can be travelled through at 5km/h.

    I started by equating the times of the 2 routes to take but came up with the wrong answer.

    Any help is greatly appreciated.

    The graphic shows 2 km at the bottom and x km on the right side.

    Find shortest time to travel-triangle.jpg
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  2. #2
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    Re: Find shortest time to travel

    Hello, ineedhelplz!

    The diagram and description are not correct.
    I think I've corrected them,


    Find the value of x (on the right vertical side) to find the minimum time to travel to the top right corner of the "forest".
    The forest can be travelled through at 3 km/h and the "clear" area can be travelled through at 5 km/h.

    The graphic shows 2 km at the bottom and x km on the right side.

    Code:
                            * B -
                            |   :
                            |d-x:
                            |   :
                            *P  :
                         *  |   d
                     *      |   :
                   *        |x  :
                *           |   :
             *              |   :
        A * - - - - - - - - *C  -
          : - - -  2  - - - :
    A man wants to get from A to B in the least amount of time.
    . . AC = 2,\;BC = d

    He walks through the "forest" from point A to point P
    . . then walks in the "clear" from point P to point B.
    Let x = CP;\;d-x = PB.


    He walks distance AP = \sqrt{x^2+4} at 3 km/hr.
    . . This takes \frac{\sqrt{x^2+4}}{3} hours.

    He walks distance PB = d-x at 5 km/hr.
    ., . This takes \frac{d-x}{5} hours.

    The total time is: . T \:=\:\tfrac{1}{3}(x^2+4)^{\frac{1}{2}} + \tfrac{d}{5} - \tfrac{1}{5}x

    And that is the function we must minimize.,
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  3. #3
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    Re: Find shortest time to travel

    Thank you very much, that makes a lot of sense, however; how do I factor the extra distance d into the equation?

    I get an answer of x=1.5 which is what I got initially. The answer in the solutions is 1.2km.

    Could you please further complete this example? That would be a great help!

    Thank you
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