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Minimizing Surface Area

  1. Mar 26, 2008 #1
    [SOLVED] Minimizing Surface Area

    1. The problem statement, all variables and given/known data
    A can is to be manufactured in the shape of a circular cylinder with volume = 50.
    Find the dimensions of a can that would minimize the amount of material needed to make the can.


    2. Relevant equations
    V = [tex] \pi r^2 h[/tex]
    SA = [tex] 2 \pi r^2 + 2 \pi r h [/tex]


    3. The attempt at a solution
    I have never done a problem like this so I am unsure how to do it, but here is my attempt.

    With the volume equation I solved for h. [tex] h = \frac{50}{\pi r^2} [/tex]
    I plugged this value for h into the Surface area equation. [tex] SA = 2 \pi r^2 + 2 \pi r \frac{50}{\pi r^2} [/tex]

    which = [tex] 2 \pi r^2 + \frac{100}{r} [/tex]

    I then took the derivative of that and set it equal to 0.
    [tex] 0 = 4 \pi r - 100 r^-2 [/tex]
    [tex] r = \sqrt[3]{\frac{100}{4 \pi}} [/tex]
    r = 1.996

    Then I plugged that back into the volume equation to solve for h and got h= 3.99.
    Could someone tell if this is right and if not where I went wrong. Thanks.
     
  2. jcsd
  3. Mar 26, 2008 #2
    Looks good to me. Check your result by plugging in a number smaller than r and a number bigger than r into the derivative. If a number smaller than r makes the derivative negative and a number larger than r makes the derivative positive, then r would be a minimum.
     
  4. Mar 26, 2008 #3
    So is that how I know that its a minimum instead of a maximum. I'm still a little confused about that. What would I do if I wanted to solve this problem for a maximum?
     
  5. Mar 27, 2008 #4

    HallsofIvy

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    There is no maximum.
     
  6. Mar 27, 2008 #5
    Does it not have one? Even though the volume is set to 50 there is no max Surface area?
     
  7. Mar 27, 2008 #6
    If you keep increasing r, the surface area just gets bigger and bigger without bound. The height gets smaller, but there is a separate [tex]2\pi r^2[/tex] term in the surface area calculation. That's why there is no maximum surface area.
     
  8. Mar 27, 2008 #7
    I understand I think. h could get infinitely small which would make the surface area infinitely large.
     
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