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  1. #1
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    optimization problem

    A cardboard box without a lid is to have a volume of 32,000 cm^3. Find the dimensions that minimize the amount of cardboard used.
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  2. #2
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    Hello, jlt1209!

    This requires partial derivatives . . .


    A cardboard box without a lid is to have a volume of 32,000 cm³.
    Find the dimensions that minimize the amount of cardboard used.
    Code:
             *- - - -*
            /|      /|
           / |     / | z
          * - - - *  |
          |       |  *
        z |       | / y
          |       |/
          * - - - *
              x

    The length, width, height of the box are: . x,y,z, respectively.

    The volume is 32,000 cm³: . xyz \:=\:32,\!000 \quad\Rightarrow\quad z \:=\:\frac{32,\!000}{xy} .[1]

    The total surface area of the box is: . A \;=\;xy + 2xz + 2yz .[2]

    Substitute [1] into [2]: . A \;=\;xy + 2x\left(\frac{32,\!000}{xy}\right) + 2y\left(\frac{32,\!000}{xy}\right)

    . . and we have: . A \;=\;xy + 64,\!000y^{-1} + 64,\!000x^{-1}


    Set the partial derivatives equal to 0.

    . . \begin{array}{cccc}<br />
\dfrac{\partial A}{\partial x} \;=\;y - 64,\!000x^{-2} \;=\;0 & \Longrightarrow & y \:=\:\dfrac{64,\!000}{x^2} & {\color{blue}[3]}\\ \\[-3mm] \dfrac{\partial A}{\partial y} \;=\;x - 64,\!000y^{-2} \;=\;0 & \Longrightarrow & x \:=\:\dfrac{64,\!000}{y^2} & {\color{blue}[4]}<br />
\end{array}

    Substitute [3] into [4]: . x \:=\:\frac{64,\!000}{\frac{64,000^2}{x^4}} \quad\Rightarrow\quad x \:=\:\frac{x^4}{64,\!000}

    . . x^4 - 64,\!000x\:=\:0 \quad\Rightarrow\quad x(x^3-64,\!000) \:=\:0

    . . x^3 \:=\:64,\!000 \quad\Rightarrow\quad\boxed{ x \:=\:40}

    Substitute into [3]: . y \:=\:\frac{64,\!000}{40^2} \quad\Rightarrow\quad\boxed{ y \:=\:40}

    Substitute into [1]: . z \:=\:\frac{32,\!000}{40\cdot40} \quad\Rightarrow\quad\boxed{ z \:=\:20}


    Therefore: . \begin{Bmatrix}\text{Length} &=& 40\text{ cm} \\ \text{Width} &=& 40\text{ cm} \\ \text{Height} &=& 20\text{ cm} \end{Bmatrix}

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  3. #3
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    Quote Originally Posted by jlt1209 View Post
    A cardboard box without a lid is to have a volume of 32,000 cm^3. Find the dimensions that minimize the amount of cardboard used.
    Are you familiar with the method of Lagrange multipliers?

    L = xy + 2yz + 2zx + \lambda (xyz - 32, 000) and ppplication of this method leads to the following equations that need to be solved simultaneously:

    0 = y + 2z + \lambda y z .... (1)

    0 = x + 2z + \lambda x z .... (2)

    0 = 2y + 2x + \lambda x y .... (3)

    32,000 = xyz .... (4)
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