A manufacturing firm produces two different products from two main raw materials. Producing one unit of product $1$ requires $2$ kg of row material $1$ and $1$kg of row material $2.$ One unit of product $2$ requires $3$ kg of row material $1$ and $2$kg of row material $2.$ The firm currently has $12$ kg of row material $1$ and $8$ kg of row material $2$ available in the stock for the next week. According to the market study, the firm estimates the weekly demand for product $1$ to be not more than $5$ units, while no restriction related to amount to be produced about product $2.$ Formulate a linear programming model that maximizes the profit given that the firm gains a marginal profit of $$2$ per unit of product $1$ and $$3$ per unit of product $2.$ a$)$ Solve the model you obtained in qusetion $2.$ Deaw the constraint lines in x y plane $.$ B$)$ define the directions of each constraint. C$)$ shade the feasible region. d$)$ find the optimal point and associated profit