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Math Help - Max/Min word problem

  1. #1
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    Max/Min word problem

    Hello everyone, I am a little stuck on this problem:

    A man is floating in a rowboat 1 mile from the straight shoreline of a large lake. A town is located on the shoreline 1 mile from the point on the shoreline closest to the man. He intends to row in a straight line to some point P on the shoreline and then walk the remaining distance to the town. To what point should he row in order to reach his destination in the least time if he can walk 5 mi/hr and row 3 mi/ hr?

    Any tips will be appreciated
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  2. #2
    Super Member

    Joined
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    Lexington, MA (USA)
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    Hello, Csou090490!


    A man is floating in a rowboat 1 mile from the straight shoreline of a large lake.
    A town is located on the shoreline 1 mile from the point on the shoreline closest to the man.
    He intends to row in a straight line to some point P on the shoreline
    and then walk the remaining distance to the town.
    To what point should he row in order to reach his destination
    in the least time if he can walk 5 mph and row 3 mph?
    Code:
          M
          *
          |  *
          |     *     ____
        1 |        * √x+1
          |           *
          |              *
          * - - - - - - - - * - - - - - *
          A       x         P    1-x    T

    The man is at M; the nearest point on shore is A.
    The town is at T: .AT = 1

    The man will row to point P and walk to point T.
    Let x = AP.
    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . _____
    From Pythagorus, the distance he rows is: .MP .= .√x + 1
    The distance he walks is: .PT .= .1 - x

    We will use: .Time .= .Distance Speed

    He rows (x + 1)^
    miles at 3 mph. .His rowing time is: .(x + 1)^/3

    He walks (1 - x) miles at 5 mph. .His walking time is: .(1 - x)/5


    His total time is: . T .= .(1/3)(x + 1)^
    + (1/5) - (1/5)x

    And that is the function we must minimize . . .

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