Maximization subject equality constraint

In summary, using Lagrange, the equations 2xyz³/50 = x²z³/10 = 3x²yz²/100 can be derived from the given function f = x²yz³ and the constraint 50x+10y+100z = 1000. After solving for the four variables, it was found that z = 5, x = 20/3, and y = 50/3. However, upon checking if these solutions satisfy the four equations and the constraint, it was discovered that there was a mistake in the calculations. After correcting the mistake, the correct solution was obtained.
  • #1
oswald
22
0

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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  • #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.
 
  • #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
 
  • #4
It's always a good idea to double-check your calculations if the result appears to be wrong. :smile:
 

1. What is a "Maximization subject equality constraint"?

A Maximization subject equality constraint is a mathematical optimization problem in which the objective is to maximize a certain function while also satisfying a set of equality constraints.

2. How is a Maximization subject equality constraint problem solved?

A Maximization subject equality constraint problem is typically solved using mathematical techniques such as Lagrange multipliers or the KKT conditions.

3. What is the difference between equality constraints and inequality constraints?

Equality constraints require that the values of certain variables be equal to a specified value, while inequality constraints only require that the values be greater than or less than a specified value.

4. Can a Maximization subject equality constraint problem have multiple solutions?

Yes, a Maximization subject equality constraint problem can have multiple solutions if the objective function and constraints are non-linear. In this case, there can be multiple optimal points that satisfy the constraints and have the same objective function value.

5. What are some real-world applications of Maximization subject equality constraint problems?

Maximization subject equality constraint problems are commonly used in economics, engineering, and business to optimize resource allocation, production processes, and portfolio management. They can also be applied in transportation planning, scheduling, and logistics.

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