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Math Help - Lagrange multipliers problem

  1. #1
    MHF Contributor arbolis's Avatar
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    Lagrange multipliers problem

    We want to construct a right prism (a 3 dimensional rectangular box) which doesn't have a cap. The material in which is constructed its base costs 5 times the material its walls are made of. (cost is per unit of area).
    I have to find the dimensions of the right prism that minimize its cost, considering that its volume must be V.
    My attempt : I've the constraint xyz=V. I'm not really sure about the function I have to minimize. I've thought about 5xz+2yz since the price of an area unit of xy=5 times the area unit of xz and yz.
    So I'd have to work out the critical points of 5xz+2yz+\lambda (xyz-V) but I found a contradiction by doing so ( x=\frac{x}{2} \Rightarrow x=0, z=+\infty).
    So I know I made an error by chosing the function f(x,y,z)=5xz+2yz. I'd like a bit of help to chose the right function to minimize. Thanks in advance.
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  2. #2
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    Hello, arbolis!

    Your surface area is off.

    I assume you labeled your box like this:
    Code:
             *---------*
            /|        /|
           / |       / |
          *---------*  |
          |         |  *
        y |         | /
          |         |/ z
          *---------*
              x

    There are four side panels.
    . . Front/back: . A_1 \:=\:2xy
    . . Left/Right: . A_2 \:=\:2yz

    There is one base panel.
    . . Bottom: . A_3 \:=\:xz

    The cost function is: . C \:=\:5xz + 2xy + 2yz


    Give it another try . . .

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  3. #3
    MHF Contributor arbolis's Avatar
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    Wow, thank you very much Soroban and yes the box is exactly the one you described. I'll give my try tomorrow. Any problem I have I'll ask help here, but I think I won't need it. Thanks a lot.
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  4. #4
    MHF Contributor arbolis's Avatar
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    I've done the problem maybe yesterday. I got x=y=\sqrt[3]{\frac{2V}{5}} while z=\frac{5}{2} \cdot \sqrt[3]{\frac{2V}{5}}. I know I can simplify the final result but I like it like that.
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