the production function is f(L, M)= 4L^1/2 M^1/2, where L is the number of units of labor and M is the number of machines used.If the cost of labor is $9 per unit and the cost of machines is $81 per unit, then the total cost of producing 10 units of output will be A)$270. B)$90. C)$135. D)$450. E)None of the above.

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Quizplus · Accepted Answer

The total cost of production is the sum of the cost of labor and the cost of machines used: Cost of labor: $9 per unit of labor * L units of labor Cost of machines: $81 per unit of machines * M units of machines We need to find the values of L and M that minimize the total cost of production, subject to the constraint that 10 units of output are produced. We can solve for L and M using the Lagrange multiplier method: minimize C = 9L + 81M subject to f(L, M) = 10 The Lagrangian function is: L = 9L + 81M + λ(10 - f(L, M)) Taking partial derivatives of L with respect to L and M, and setting them equal to 0, we get: ∂L/∂L = 9 - λF1^1/2 = 0 ∂L/∂M = 81 - λF1^1/2 = 0 We also need to take the partial derivative of L with respect to λ: ∂L/∂λ = 10 - f(L, M) = 0 Solving these equations simultaneously, we get: F1^1/2 = 1/9 81/9 = F1^1/2 F1^1/2 = 9 Substituting F1^1/2 = 9 into the production function, we get: f(L, M) = 10 = 4L^1/2 * 9M^1/2 Simplifying, we get: L^(1/2)M^(1/2) = 10/36 We want to find the values of L and M that satisfy this equation and minimize the cost function. Substituting L^(1/2)M^(1/2) = 10/36 into the cost function, we get: C = 9L + 81M = 9(36L^(1/2)M^(1/2)) + 81M = 324(LM)^(1/2) + 81M Substituting LM = (10/36)^2, we get: C = 324(10/36) + 81(10/36) = $90 Therefore, the total cost of producing 10 units of output is $90, and the best choice is (C).

The Production Function Is F(L, M)= 4L1/2 M1/2, Where L

The Production Function Is F(L, M)= 4L^1/2 M^1/2, Where L