Maximization subject equality constraint

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Homework Help Overview

The problem involves maximizing the function f = x²yz³ subject to the constraint 50x + 10y + 100z = 1000, using Lagrange multipliers.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the use of Lagrange multipliers and the resulting equations derived from the method. There are attempts to verify the correctness of the solutions found and to check if they satisfy the original equations and constraints.

Discussion Status

Some participants have provided guidance on verifying the solutions against the equations. There is acknowledgment of a mistake in the calculations, and further exploration of the equations is ongoing.

Contextual Notes

Participants are encouraged to show their work in solving the equations to facilitate assistance. There is an emphasis on ensuring all equations are satisfied with the proposed solutions.

oswald
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Homework Statement


Max f = x²yz³
sub. 50x+10y+100z = 1000


Homework 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, what's wrong?

The Attempt at a Solution



z = 5
x = 20/3
y = 50/3
 
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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.
 
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
 
It's always a good idea to double-check your calculations if the result appears to be wrong. :smile:
 

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