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Math Help - Calc (Optimization)

  1. #1
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    Calc (Optimization)



    Heres what I did so far:

    Plug 3c in for k:

    S = c/x^2 + 3c/(d-x)^2

    I differentiate, set equal to 0, and solve for x to get:

    0=-c(2x)+-3c[2(d-x)(-1)]
    0=-2cx+6cd-3cx
    0=-5cx+6cd
    x=6d/5

    x min = 6d/5 miles

    However this is not the correct answer.... Any help would be greatly appreciated. Thanks!
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  2. #2
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    Quote Originally Posted by pakman134 View Post


    Heres what I did so far:

    Plug 3c in for k:

    S = c/x^2 + 3c/(d-x)^2

    I differentiate, set equal to 0, and solve for x to get:

    0=-c(2x)+-3c[2(d-x)(-1)]
    0=-2cx+6cd-3cx
    0=-5cx+6cd
    x=6d/5



    x min = 6d/5 miles

    However this is not the correct answer.... Any help would be greatly appreciated. Thanks!
    You mess up on the derivative

    S = c/x^2 + 3c/(d-x)^2=cx^{-2}+3c(d-x)^{-2}

    \frac{dS}{dx}=-2cx^{-3}-6c(d-x)^{-3}(-1)=\frac{-2c}{x^3}+\frac{6c}{(d-x)^3}

    Good luck.

    B
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  3. #3
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    Quote Originally Posted by TheEmptySet View Post
    You mess up on the derivative

    S = c/x^2 + 3c/(d-x)^2=cx^{-2}+3c(d-x)^{-2}

    \frac{dS}{dx}=-2cx^{-3}-6c(d-x)^{-3}(-1)=\frac{-2c}{x^3}+\frac{6c}{(d-x)^3}

    Good luck.

    B
    hm.. I know this almost sounds like I'm asking you to do the problem for me... But I'm a little lost as to how you go about solving for x now.... =p
    I tried playing around with it and factoring out the (d-x)^3
    0 = -2c/x^3 + 6c/(d-x)^3
    0 = (-2c/x^3) + 6c/(d^3-3d^2x+3dx^3-x^3)

    And from here I'm lost once again... Do i set a lowest common denom now or am I completely off track...
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  4. #4
    Behold, the power of SARDINES!
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    no worries

    <br />
\frac{dS}{dx}=-2cx^{-3}-6c(d-x)^{-3}(-1)=\frac{-2c}{x^3}+\frac{6c}{(d-x)^3}<br />

    multiply by LCD x^3(d-x)^3

    0=-2c(d-x)^3+6cx^3=-2c[(d-x)^3-3x^3]

    Note that A^3-B^3=(A-B)(A^2+AB+B^2)

    rewriting we get...

    -2c[(d-x)^3-(\sqrt[3]{3}x)^3]

    With A=(d-x) \mbox{ and } B=(\sqrt[3]{3}x)

    So we get...

    0=-2c[(d-x)-\sqrt[3]{3}x][(d-x)^2+(d-x)((\sqrt[3]{3}x)+((\sqrt[3]{3}x)^2]

    Setting the 1st factor equal to zero we get...

    (d-x)-\sqrt[3]{3}x solving for x we get

    d=x+\sqrt[3]{3}x \iff d=(1+\sqrt[3]{3})x \iff x=\frac{d}{1+\sqrt[3]{3}} \approx 0.40947d
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