A city of $15,000$ habitants requires $300,000$ gallons of water per day. The city is supplied with water from a central processing plant, where the water is filtered, chlorinated, and enriched with chemicals X and Y$.$ The company operating the plant plans to purchase two product brands A and B containing the following: One unit of product A provides $8$kg of X and $3$kg of Y$,$ while one unit of product B provides $4$kg of X and $9$kg of Y$.$ To maintain water quality standards, it is specified that at least $150$ kg total of chemical X and $100$ kg of chemical Y are required daily. If the costs associated with A and B are $$8$ and $$10$ respectively, what is the optimal daily amount of each product to add?

A$)$ Formulate the primal model and solve it$.$ What are the basic variables, their values, and the value of the objective function?

B$)$ Formulate the dual model. What is the opportunity cost for each constraint?

C$)$ Perform a sensitivity analysis: Within what range can the requirement for chemical Y be modified without changing the optimal solution?

D$)$ Perform a sensitivity analysis: Within what range can the price of product A be modified without changing the optimal solution?