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Math Help - For what point along the shoreline should you aim?

  1. #1
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    For what point along the shoreline should you aim?

    You are in a row boat on Lake Erie, 2 miles from a straight shoreline, taking your potential in-laws for a boat ride. Six miles down the shoreline from the nearest point on shore is an outhouse. You suddenly feel the need for its use. Also, the shore line is populated with lots of houses, all owned by people who know your parents. If you can row at 2 mph and run at 6 mph, for what point along the shoreline should you aim in order to minimize the amount of time it will take you to get to the outhouse?
    Ok, I drew a picture of it. I am guessing that I should first row the boat diagonally to a point instead of rowing vertically down to the shore, because rowing diagonally will shorten the distance. Please I need someone to help me figuring out this problem.


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  2. #2
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    Set the closest landing point from the boat as A(it will be directly perpendicular from the shore to the boat )
    Set an assumed landing point you have chosen as B
    Assign the distance between A and B as x
    For whatever value of x, the distance you have to row is
    sqr root of ( 2^2 + x^2) miles
    (Pythagoras theorem. I hope you’ve studied)
    Therefore time taken to row will be
    sqr root of ( 2^2 + x^2) miles /2 mph
    =sqr of ( 2^2 + x^2)/2 hrs
    ( time = distance / speed )
    Next you find the time you use to run from where you have landed to the house
    The remaining distance to run is
    6-x miles
    And the time used is
    6-x miles ÷ 6mph = 6-x/6hours
    ( note that speed = distance /time
    So time = distance / speed)
    Now you’ve got the eqn:
    time to row + time to run
    = sqr of ( 2^2 + x^2)/2 hrs+ 6-x/6hours
    =sqr of( 2^2 + x^2)/2 + 6-x/6 hours
    THIS will be the function of your graph of time against distance of landing from boathouse
    Y = sqr of (2^2 + x^2)/2 + 6-x/6
    In which Y is time and x is distance between point A and B ( refer drawing )
    Now differentiate the graph to find its minimum point
    I hope you've understood. too bad I can't draw anything in this stupid replying column. e mail me for a better description.
    You should have studied this when you were in secondary school...
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  3. #3
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    Very clear~ thank you very much!
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  4. #4
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    For what point along the shoreline should you aim?-boat.bmp
    that's what i meant
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  5. #5
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    solution1.doc
    here's the whole thing
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