Connor owns a model car company that produces miniature race cars and muscle cars that are moulded from aluminium sheets and finished by hand. Each race car is made from 5 pounds of material, requires 4 minutes of machine time and 2 hours of finishing labour, and can be sold for $150. Each muscle car is made from 6 pounds of material, requires 5 minutes of machine time and 3 hours of finishing labour, and can be sold for $250. Each month, the production team has access to 8 hours of machine time, 200 hours of finishing labour and 200 pounds of aluminium. His maximum revenue is $8333 and that is with 0 racecars and 33.33 muscle cars.

Sarah is an investor and offers Connor a business arrangement. She wants to operate the business under a lease agreement; she will pay him P dollars for each pound of aluminum she uses, M dollars for each minute of machine time she uses and H dollars for each hour of finishing labour she uses. She guarantees him that her lease payment will at least cover the income he would have had by doing the work himself.

(a) Considering separately the revenues for race cars and muscle cars, construct 2 inequalities that P, M and H must satisfy. Write a third expression of P, M and H for the maximum total amount Sarah will pay, that is, in the case that she uses up all the available resources.

(b) Put the Question 1 problem in standard form for finding the dual and use it to find the dual problem according to the method seen in class (Slide 105). What do you notice, compared to your results in 2(a)? Explain the physical interpretation of the dual problem formulas in this case.

(c) Without solving the dual problem, write its solution (optimizer and optimal cost) and explain how you obtained it.