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Math Help - Optimization Problem2

  1. #1
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    Optimization Problem2

    A box with an open top is to be constructed from a square piece of cardboard, 3ft wide, by cutting out a square from each four corners and bending up the sides. Find the largest volume that such a box can have.
    Draw several diagram and write that relates the variables.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by camherokid View Post
    A box with an open top is to be constructed from a square piece of cardboard, 3ft wide, by cutting out a square from each four corners and bending up the sides. Find the largest volume that such a box can have.
    Draw several diagram and write that relates the variables.
    see diagram below.

    Let the sides of the squares cut out be x

    then V = lwh
    => V = (3 - 2x)(3 - 2x)x
    => V = (3 - 2x)^2*x
    => V = (9 - 12x + 4x^2)x
    => V = 4x^3 - 12x^2 + 9x

    => V' = 12x^2 - 24x + 9

    for max V, set V' = 0
    => 12x^2 - 24x + 9 = 0
    => 4x^2 - 12x + 3 = 0
    => (2x - 3)(2x - 1) = 0
    => 2x - 3 = 0 or 2x - 1 = 0
    => x = 3/2 or x = 1/2

    now V'' = 24x - 24
    when x = 3/2
    V'' = 24(3/2) - 24 = 12 > 0
    this is a concave up, so this is a local min, so this is the minimum value based on the second derivative test

    when x = 1/2
    V'' = 24(1/2) - 24 = -12 < 0
    this is a concave down, so this is a local max, so this is the maximum value based on the second derivative test

    so for the max Volume, which we will call maxV, x = 1/2

    but V = (3 - 2x)^2 * x
    => maxV = (3 - 2(1/2))^2 * (1/2)
    => maxV = 2 cm^2


    EDIT: oh, sorry, we're working in ft, just change all the cm to ft in the diagram and my calculations
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