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Math Help - Optimization

  1. #1
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    Optimization

    Mary is standing at the edge of a slow-moving river which is one mile wide and she wishes to return to her campground site on the opposite side of the river. She can swim at a rate of 2 mph and walk at a rate of 3 mph. She must first swim across the river to any point on the opposite bank. From there she can walk to the campground, which is one mile down river from the point directly across the river from where she start her swim. What route will take the least amount of time?

    What I started to do was find the missing leg of the triangle which is √2 because i figured that would be the quickest route across, but I dont know what to do after that any ideas? Thanks.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    On my website is a similar problem -

    See a bird flies home at

    Optimization and Related Rates

    This should give you an idea how to proced --only the numbers are different
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  3. #3
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    According to your bird example, am I on the right track with this pic:

    Its sort of small but its basically x distance walking, then √(1+(1-x)^2) distance across to the campsite. It forms a triangle with base 1-x, height 1 and hypotenuse √(1+(1-x)^2) ? So from that, my time equation would look like:

    t(x)= x/3+√(1+(1-x)^2) / 2
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  4. #4
    MHF Contributor Calculus26's Avatar
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    the equation is correct but the way the problem is stated she must swim first then walk.

    In your diagram she walks first and swims second

    if you switch the campground and Mary the diagram matches the question
    and the equation stays the same
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  5. #5
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    Thanks for the help, much appreciated!
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