A firm uses two groups of workers, machinists X₁ and electrical engineers X₂, to process components for computers. Machinists are paid $10 per hour and electrical engineers are paid $20 per hour.

The production function P expressing processed components per hour as a function of the number of workers in the two groups is:

P = 2X₁² + 3X₂² + 30X₁ + 40X₂

The firm wants to spend exactly $410 per hour on labor costs related to production of these components. Within this constraint, the firm wants to use a mix of workers of the two categories which will maximize the number of components processed per hour.

a.Set up the Lagrange function Z for this problem of constrained maximization.

b.How many employees of each group should be used to maximize hourly output within the constraint?

c.What is the number of components manufactured per hour using the amounts of labor found in part b?