Results 1 to 2 of 2

Math Help - Lagrange Multipliers

  1. #1
    Newbie
    Joined
    Mar 2008
    Posts
    5

    Lagrange Multipliers

    A manufacturing company has three plants, I, II and III, which produce x, y, and z units, respectively, of a
    certain product. The annual revenue from this production is given by
    R(x, y, z) = 6xyz^2 − 400000(x + y + z)
    If the company is to produce 1000 units anually, use Lagrange Multipliers to determine how it should allocate
    production among the three plants so as to maximize its revenues.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    We have f(x,y,z)=6xyz^{2}-400,000(x+y+z) subject to the constraint g(x,y,z)=x+y+z-1000[/tex]

    {\nabla}f(x,y,z)={\lambda}{\nabla}g(x,y,z)

    f_{x}=6yz^{2}-400,000={\lambda}....[1]

    f_{y}=6xz^{2}-400,000={\lambda}...[2]

    f_{z}=12xyz-400,000={\lambda}...[3]

    Equate [1] and [2]:

    6yz^{2}=6xz^{2}; \;\ x=y

    Equate [1] and [3]:

    6yz^{2}-400,000=12xyz-400,000

    z=2x

    Sub into the constraint:

    x+x+2x=1000

    4x=1000

    \boxed{x=250, \;\ y=250, \;\ z=500}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. lagrange multipliers
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 10th 2009, 03:24 AM
  2. Lagrange Multipliers
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 1st 2009, 12:49 PM
  3. Lagrange Multipliers in 4D
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 23rd 2009, 10:37 AM
  4. Lagrange multipliers
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 22nd 2009, 02:17 PM
  5. Lagrange Multipliers
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 2nd 2009, 10:26 PM

Search Tags


/mathhelpforum @mathhelpforum