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Math Help - Optimization and Differentials

  1. #1
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    Optimization and Differentials

    Hi...I'm working on a problem for the minimization of materials for a can. This juice can should hold 163 mL, and I'm already stuck on the first step. When setting up the minimization equation I should be able to graph it and get an idea of about where the minimum should be. I've got 2*Pi*r^2 + 326/r so far, but the graph doesn't show anything that looks like a zero slope anywhere - it just looks like a standard sqrt function. So before I go further with this, what am I doing wrong?
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by marqade View Post
    Hi...I'm working on a problem for the minimization of materials for a can. This juice can should hold 163 mL, and I'm already stuck on the first step. When setting up the minimization equation I should be able to graph it and get an idea of about where the minimum should be. I've got 2*Pi*r^2 + 326/r so far, but the graph doesn't show anything that looks like a zero slope anywhere - it just looks like a standard sqrt function. So before I go further with this, what am I doing wrong?
    Did you start with 163=\pi{r^2}h

    and S=2\pi{r}(r+h)?
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  3. #3
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    I started with 163 = Pi*r^2*h;
    A = 2*Pi*r^2 + 2*Pi*r*h;
    h = 163/(Pi*r^2)
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by marqade View Post
    I started with 163 = Pi*r^2*h;
    A = 2*Pi*r^2 + 2*Pi*r*h;
    h = 163/(Pi*r^2)
    Thats correct...I just factored by suface area

    Now imput that into yoru surface area to get

    S=2\pi{r}\bigg(r+\frac{163}{\pi{r^2}}\bigg)

    This means that S'=2\pi{r}\bigg(1-\frac{326}{\pi{r^3}}\bigg)+2\pi\bigg(r+\frac{163}{  \pi{r^2}}\bigg)

    Now solve S'=0
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  5. #5
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    If we cut this same can from hexagons instead, thereby saving even more material, could I write the area as a function of the radius as such: A(r) = 6*(1/2)(2/r)(r)? because 1/2 b*h = A; b = 2/r and the height = radius. Problem with this is that my variable cancel themselves out and I'm not sure what to do with that. Do I need to be using trig formulas?
    x = (1/r*sin90)/(sin(45)) = sin(45)/r = (sqrt(2))/2r.
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