Find the minimum cost of producing 50,000 units of a product, where x is the number of units of labor (at $36 per unit) and y is the number of units of capital (at $48 per unit).

(Round your answer to two decimal places.)

P (x, y) = 100x^0.6y^0.4

Step-by-step explanation:

Given problem: C (x, y) = 36x + 48y

constraint: 100x^0.6y^0.4

Using langrange Multiplier,

36 = 0.6 (100) x^-0.4y^0.4λ i

48 = 0.4 (100) x^0.6y^-0.6λ ii

dividing the equations we have:

x = 2y

substituting into the constraint

p (x, y) = 100 * (2y) ^0.6 y^0.4 = 100*2^0.6 * y

5000 = 151.572y

y = 329.876 labor units

x = 659.752 capital units

Minimum cost = 36 (659.752) + 48 (329.876) = $39585.12