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Maximization subject equality constraint

  1. Apr 10, 2008 #1
    1. The problem statement, all variables and given/known data
    Max f = x²yz³
    sub. 50x+10y+100z = 1000


    2. Relevant equations

    Using Lagrange:

    L = x²yz³ - λ ( 50x + 10 y + 100z - 1000 )
    Lx = 2xyz³ - λ50 = 0
    Ly = x²z³ - λ10 = 0
    Lz = 3x²yz² - λ100 = 0

    i found z = 2,5 x= 10 y=25, whats wrong?
    3. The attempt at a solution

    z = 5
    x = 20/3
    y = 50/3
     
  2. jcsd
  3. Apr 10, 2008 #2
    Did you check if your solutions satisfy the four equations, (which appear to be right):

    Lx = Ly =Lz = 0 and the constraint 50x+10y+100z = 1000 ?

    If they do your solution is correct, if not you made a mistake in solving the four equations and you should show us how you tried to do it so that we can help you.

    Note, that you have four equations and four variables to solve for: x, y, z and lambda.
     
  4. Apr 11, 2008 #3
    2xyz³/50 =λ
    x²z³/10 =λ
    3x²yz²/100= λ
    hence,
    2xyz³/50 = x²z³/10 = 3x²yz²/100

    substituting:

    2xyz³/50 = x²z³/10
    2y/50 = x /10
    20 y = 50x

    x²z³/10 = 3x²yz²/100
    z/10 = 3y/100
    100 z = 30y

    50x+10y+100z = 1000
    20y + 10y + 30y = 1000
    60y = 1000
    y = 50/3
    ah, i made a mistake in solving the four equations... thanks
     
  5. Apr 11, 2008 #4
    It's always a good idea to double-check your calculations if the result appears to be wrong. :smile:
     
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