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Thread: optimization problem

  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: .$\displaystyle x,y,z$, respectively.

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

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

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

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


    Set the partial derivatives equal to 0.

    . . $\displaystyle \begin{array}{cccc}
    \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]}
    \end{array} $

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

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

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

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

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


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

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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?

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

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

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

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

    $\displaystyle 32,000 = xyz$ .... (4)
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