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Math Help - Substition Within an Optimization Problem

  1. #1
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    Substition Within an Optimization Problem

    I am working on an optimization problem right now and ran into the following situation.

    V = piR^2H = 8
    H = 8/piR^2
    H = 2srt(2)/piR

    A = H2piR+piR^2
    A = (2rt(2)/piR)2piR+piR^2
    A = 4srt(2)+piR^2

    dA/dR = 2piR

    I think I have done this wrong, but I am fairly tired right now and can't seem to find my mistake.
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  2. #2
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    Quote Originally Posted by Vanilla View Post
    I am working on an optimization problem right now and ran into the following situation.

    V = piR^2H = 8
    H = 8/piR^2
    H = 2srt(2)/piR incorrect

    A = H2piR+piR^2
    A = (2rt(2)/piR)2piR+piR^2
    A = 4srt(2)+piR^2

    dA/dR = 2piR

    I think I have done this wrong, but I am fairly tired right now and can't seem to find my mistake.
    Are you trying to minimise the surface area of the cylinder for a fixed volume of 8 cubic units?
    Last edited by Archie Meade; October 11th 2010 at 03:21 PM.
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  3. #3
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    Not sure where I got the idea that [LaTeX ERROR: Convert failed] .

    Quote Originally Posted by Archie Meade View Post
    Are you trying to maximise the surface area of the cylinder for a fixed volume of 8 cubic units?
    Trying to minimize the surface area. However; I know how to solve the rest of the problem. Thanks for the help.

    Edit: Actually, I got stuck again.

    [LaTeX ERROR: Convert failed] [LaTeX ERROR: Convert failed]
    [LaTeX ERROR: Convert failed]
    [LaTeX ERROR: Convert failed]
    [LaTeX ERROR: Convert failed]
    [LaTeX ERROR: Convert failed]

    Then solving for dA/dR = 0.

    [LaTeX ERROR: Convert failed]
    [LaTeX ERROR: Convert failed]
    [LaTeX ERROR: Convert failed]

    And from there I can't find a way to isolate R, probably because I made some silly algebraic error.
    Last edited by Vanilla; October 11th 2010 at 03:04 PM.
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  4. #4
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    Hi Vanilla,

    The volume of a cylinder is {\pi}R^2H

    The curved surface area is 2{\pi}RH

    The flat surface area is {\pi}R^2

    The total surface area of a cylinder closed at both ends (surface area of a solid cylinder) is

    2{\pi}RH+2{\pi}R^2

    V={\pi}R^2H=8\Rightarrow\ H=\displaystyle\frac{8}{{\pi}R^2}

    Substitute the equation for H into A to get an equation for SA in terms of the single variable R...

    Surface area =\displaystyle\ 2{\pi}R\ \frac{8}{{\pi}R^2}+2{\pi}R^2=\frac{16}{R}+2{\pi}R^  2

    Differentiating with respect to R and setting derivative to zero
    finds the value of R that gives minimum surface area.

    Double check whether or not the cylinder is closed.
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  5. #5
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    The cylinder is open on one end. So the [LaTeX ERROR: Convert failed] in the surface area equation would instead be {\pi}R^2, correct?
    So differentiating [LaTeX ERROR: Convert failed] would still leave me with [LaTeX ERROR: Convert failed] for which I can't seem to isolate R when solving for 0.
    Last edited by Vanilla; October 11th 2010 at 04:13 PM.
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  6. #6
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    Yes,

    you only need to adjust for that before calculating the derivative.
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  7. #7
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    Double post.
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  8. #8
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    I think I got it.
    [LaTeX ERROR: Convert failed] can be simpified to [LaTeX ERROR: Convert failed] , which then differentiates to [LaTeX ERROR: Convert failed] . And for [LaTeX ERROR: Convert failed] , [LaTeX ERROR: Convert failed] . Thanks for the help!
    Last edited by Vanilla; October 11th 2010 at 04:37 PM.
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  9. #9
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    Quote Originally Posted by Vanilla View Post
    I think I got it.
    [LaTeX ERROR: Convert failed] can be simpified to [LaTeX ERROR: Convert failed] , which then differentiates to [LaTeX ERROR: Convert failed] . And for [LaTeX ERROR: Convert failed] , [LaTeX ERROR: Convert failed] . Thanks for the help!
    Yes,

    but as the numerator is zero for R giving minimum area, you need to take the cube root
    rather than the square root.
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