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Math Help - Optimization Problem Need Almost Done Stuck

  1. #1
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    Smile Optimization Problem Need Almost Done Stuck

    Hi I am stuck on a optimization problem so I am here for some assistance. Heres the question:

    Simon Suffers an injury at a campsite on the bank of a canal that is 6√2 km wide. If he is able to paddle his kayak at 5 km/h and pedal a bicycle at 15 km/h along the opposite bank, where should he land on the opposite bank to reach the hospital in minimum time?

    here is a diagram that comes with the question:
    Optimization Problem Need Almost Done Stuck-optimizationquestion.png

    So out of this question I've determined this:
    Distance = Velocity * time <- this can be re-arranged to this -> time = Distance / Velocity

    distance on kayak: √((6√2)^2 + x^2)
    distance on land: (4 - x)

    so my formula is this:
    Time(x) = (1/15)(4 - x) + (1/5)(
    √(x^2 + 72))

    I differentiated and got this:
    Time'(x) = [ (x) / (5 *
    √(x^2 + 72) ) ] - (1/15)

    Now the next step I do is I set my Time'(x) = 0 and solve for x but I feel that I am getting a bad answer. Heres what I have:
    t(x) = 0
    (x/(5sqrt(x^2 + 72)) (1/15) = 0
    (x/(5sqrt(x^2 + 72)) = (1/15)
    (x)^2 = (1/15)^2(5sqrt(x^2 + 72)^2
    x^2 = (1/225)(25)(x^2 + 72)
    (x^2) * 225 = ((1/225)(25)(x^2 + 72))*225
    (225x^2)/25 = ((5625)(225x^2 + 405000))/25
    9x^2 = (225)(9x^2 +16200)
    9x^2 = 2025x^2 + 3645000
    9x^2 - 2025x^2 = 3645000
    -2016x^2 - 3645000
    x^2 = -1808.35714x^2

    So I cant continue this because as you can see 3645000 does not divide nicely with -2016 and I cant take a square root of a negative number.

    If someone could point out where the mistake is that would be GREATLY !! appreciated!!!

    Thanks!!!




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  2. #2
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    Re: Optimization Problem Need Almost Done Stuck

    The calculation can be made much more simpler rather than finding a mistake in your approach to the problem.
    x/5*sqrt(x^2+72)=1/15
    15x=5*sqrt(x^2+72)
    3x=sqrt(x^2+72)
    9*x^2=x^2+72
    8*x^2=72
    x^2=9
    x=3
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  3. #3
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    Re: Optimization Problem Need Almost Done Stuck

    Ok thanks shyamnair3. I should of simplified more instead of over complicating it.

    Now the 3 I can plug into my first equation to get the minimal time to answer this question right?
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  4. #4
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    Re: Optimization Problem Need Almost Done Stuck

    Can someone please explain how the derivative of this is taken? I have been stuck on this exact question for a very very long time...

    my equation is t=(3(sqrt(x^2+72)+4-x)/15

    I need to know how to get the derivative of this. I have looked up this exact question is a few places and nobody actually shows how to derive this. Everybody ends up with an answer of x=3 and I have tried an exhausting amount of methods to try and get that.
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  5. #5
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    Re: Optimization Problem Need Almost Done Stuck

    anybody??
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  6. #6
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    Re: Optimization Problem Need Almost Done Stuck

    t=\frac{3(\sqrt{x^2+72}+4-x)}{15}=\frac{\sqrt{x^2+72}+4-x}{5}

    t=\frac{\sqrt{x^2+72}+4-x}{5}

    I'll just show how to get the derivative of \sqrt{x^2+72} the rest is straightforward
    y=\sqrt{x^2+72}

    \text{Let } u=x^2+72

     \frac{du}{dx}=2x

    y=\sqrt{u}

    \frac{dy}{du}=0.5u^{-0.5}

    \frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}

    \frac{dy}{dx}=0.5u^{-0.5}\cdot 2x

    \frac{dy}{dx}=0.5(x^2+72)^{-0.5}\cdot 2x
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  7. #7
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    Re: Optimization Problem Need Almost Done Stuck

    Thanks a lot man, I can see where I messed up now! Appreciate it!!
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