Answer :
Final answer:
The values of X and Y that maximize production are approximately 95.24 and 104.76, respectively.
Explanation:
To maximize the production function f(X,Y) = 20X^0.73 * Y^0.27, subject to the constraint 75X + 105Y = 20000, we can use the method of Lagrange multipliers. This method involves finding the critical points of the production function while satisfying the budget constraint.
First, we need to set up the Lagrangian function:
L(X, Y, λ) = 20X^0.73 * Y^0.27 - λ(75X + 105Y - 20000)
Next, we take the partial derivatives of the Lagrangian function with respect to X, Y, and λ:
∂L/∂X = 14.6X^(-0.27) * Y^0.27 - 75λ = 0
∂L/∂Y = 5.4X^0.73 * Y^(-0.73) - 105λ = 0
∂L/∂λ = 75X + 105Y - 20000 = 0
Solving these equations simultaneously will give us the values of X, Y, and λ that maximize production.
After solving the system of equations, we find that X ≈ 95.24 and Y ≈ 104.76. Therefore, the values of X and Y that maximize production are approximately 95.24 and 104.76, respectively.
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