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Math Help - Lagrange Multipliers

  1. #1
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    Red face Lagrange Multipliers

    The base of an aquarium in the shape of rectangular box with a given volume V is made of slate and the four sides are made of glass. If slate costs five times as much as glass per unit area, find the dimension of the aquarium that minimize the cost of the materials.


    help please~
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  2. #2
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    Quote Originally Posted by CandyKanro View Post
    The base of an aquarium in the shape of rectangular box with a given volume V is made of slate and the four sides are made of glass. If slate costs five times as much as glass per unit area, find the dimension of the aquarium that minimize the cost of the materials.


    help please~
    Let the dimensions of the box be x,\ y,\ z , then the cost is proportionsl to:

    f(x,y,z)=5 xy + 2xz + 2yz

    and we have the constraint:

    g(x,y,z)=xyz

    with g(x,y,z)=V.

    Then our Lagrangian is:

    \Lambda(x,y,z,\lambda)=f(x,y,z)-\lambda(g(x,y,z)-V)

    Now you need to find the stationary points of \Lambda to find the candidates for the maximising solution.

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