Solve the following problem by hand using mathematical optimization:

An automobile company organizes its vehicles into three main categories, 1) Pickup Trucks, 2) SUVs, 3) Sedans. The Sedan Family group has two main models. The first is a 4-door Family sedan, and the second is a 2-door luxury sedan. The Family sedan generates a profit of $3,800, while the luxury sedan generates a profit of $5,600 for the company. The assembly plant has a capacity of 51,000 labor hours during the month of May. It takes 7 hours to assemble one Family sedan and 11.5 hours to assemble one luxury sedan. The door supplier can produce a maximum of 24,000 doors (12,000 right and 12,000 left). Both models use the same door.

Determine the number of family sedans and luxury sedans that need to be assembled to maximize profit for May.