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'Simple' Optimization Problem

  1. Feb 14, 2012 #1
    Just reviewing for a chapter test... I've always found optimization problems easy, but I don't have answers for these review questions so I thought I'd check my work on here.

    1. The problem statement, all variables and given/known data
    A piece of cardboard is 14 inches by 10 inches and you are going to cut out the corners and fold up the sides to form an open box. Determine the height of the box that will give a maximum volume.

    2. Relevant equations

    V = lwh

    3. The attempt at a solution

    (10-2x) as l
    (14-2x) as w
    and x as the height:

    V = (10-2x)(14-2x)x = 4x^3 - 48^2 +140x

    dv/dx = 12x^2 - 96x + 140

    0 = 12x^2 - 96x +140

    x = 6.08, 1.92 or 1/3(12±√39)

    .... this is where I get lost. I feel like I messed it up somewhere because I get weird numbers for x. Technically, the 6.08 number is impossible, so x would have to equal 1.92. But still, I feel uncomfortable with these numbers.
  2. jcsd
  3. Feb 14, 2012 #2


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    Science Advisor
    Gold Member

    What makes you uncomfortable about the answer you got?

    Would it make you more comfortable to compare the volume if the height were slightly above or below the height you calculated to the volume you got?
  4. Feb 14, 2012 #3
    haha, yes, actually. It helped quite a bit. I think I'm just too tired from too many late nights of studying... thanks for the help.
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