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    You want to make an opened top box with volume 4. You are required to use as little material as possible in making it. Find the dimensions of the box so that the area of the box is minimised.

    In this question I have done the following:

    \mathrm{Area} = 2xy + 2yz + xz

    \mathrm{Volume} = xyz = 4

    Forming the following LaGrange's Equation:

     L(x,y,z) = 2xy + 2yz + xz -\lambda (xyz - 4)

    Solving it:

    L_x = 2y + z -\lambda yz
    L_y = 2x + 2z - \lambda xz
    L_z = 2y + x - \lambda xy
    L_\lambda = -xyz + 4

    Here is where the problem starts....

    How do I solve these??

    The answer is H = 1 and Square base of side 2.

    Since I cannot delete this thread.....I have solved it....made a mistake by taking 3 variables making it inevitably hard!

    I can put up the solution just for like reference to anyone who wants to have a go.
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    (Original post by mathsRus)
    You want to make an opened top box with volume 4. You are required to use as little material as possible in making it. Find the dimensions of the box so that the area of the box is minimised.

    In this question I have done the following:

    \mathrm{Area} = 2xy + 2yz + xz

    \mathrm{Volume} = xyz = 4

    Forming the following LaGrange's Equation:

     L(x,y,z) = 2xy + 2yz + xz -\lambda (xyz - 4)

    Solving it:

    L_x = 2y + z -\lambda yz
    L_y = 2x + 2z - \lambda xz
    L_z = 2y + x - \lambda xy
    L_\lambda = -xyz + 4

    Here is where the problem starts....

    How do I solve these??

    The answer is H = 1 and Square base of side 2.

    Since I cannot delete this thread.....I have solved it....made a mistake by taking 3 variables making it inevitably hard!

    I can put up the solution just for like reference to anyone who wants to have a go.
    Btw, AM-GM solves it quite quickly, definitely quicker than lagrange multipliers.

    Alternative solution:
    Spoiler:
    Show

    By AM-GM,  2xy+2yz+zx\geq(4x^2y^2z^2)^{ \frac{1}{3}}=4 from  xyz=4
    Then  2xy=2yz=zx at equality, giving 2y=x=z=2
 
 
 
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