The Glickman Electronics Company in St$.$ Thomas, produces two products: $(1)$ the Glickman x$-$pod and $(2)$ the Glickman Blueberry. The production process for each product is$.$ similar in that both require a certain number of hours of electronic work and a certain number. of labor$-$hours in the assembly department. Each x$-$pod takes $4$ hours of electronic work and $2$ hours in the assembly shop. Each Blueberry requires $3$ hours in electronics and $1$ hour in assembly. During the current production period, $240$ hours of electronic time are available, and $100$ hours of assembly department time are available. The company is limited to $60$ Blue Berries. Each x$-$pod sold yields a profit of $$7$; each Blueberry produced may be sold for a $$5$ profit. Glickman must maximise it's profits to remain competitive

a$)$ Use X$1$ and X$2$ to define the variables indicating which variable is X$1$ and X$2$ :

$2$ marks

b$)$ Derive the Objective Function in terms of X$1$ and X$2$

$3$ Marks

c$)$ Formulate the Constraints under which the company must operate

$7$ Marks

d$)$ Graphically illustrate the Feasible Region

$6$ Marks

e$)$ State the coordinates of the corner points e$.$g A$(2,7),$ B$(12,5)$

$7$ Marks

f$)$ Determine the optimal solution

$3$ Marks

g$)$ Determine the profit at the optimal solution