given π=1000x - 2x - xy + 180y - 4y. subject to profit constraints x + y =30. using solution method, what is the optimal profit under contraint

To find the optimal profit under the given constraint, we'll use the method of solving simultaneous equations called "Lagrange Multipliers."

Step 1: Set up the Lagrangian function.
The Lagrangian function combines the objective function (π) and the constraint function (x + y - 30) using Lagrange multipliers (λ).

L = π - λ(x + y - 30)

Step 2: Partial derivatives.
Take the partial derivative of L with respect to each variable (x, y, and λ) and set them equal to zero.

∂L/∂x = 1000 - 2 - λ = 0
∂L/∂y = -x + 180 - λ = 0
∂L/∂λ = x + y - 30 = 0

Step 3: Solve the equations.
Solve the system of equations obtained from the partial derivatives.

From the first equation: 1000 - 2 - λ = 0 <=> λ = 998
From the second equation: -x + 180 - λ = 0 <=> -x + 180 - 998 = 0 <=> x = 782

Substituting the value of x into the third equation: 782 + y = 30 <=> y = -752

Step 4: Calculate the optimal profit.
Substitute the values of x and y obtained from solving the equations back into the objective function (π) to calculate the optimal profit.

π = 1000x - 2x - xy + 180y - 4y
= 1000(782) - 2(782) - (782)(-752) + 180(-752) - 4(-752)
= 782000 - 1564 + 587264 + (-135360) + 3008
= 1236848

Therefore, the optimal profit, subject to the profit constraint x + y = 30, is 1,236,848.