$1)$ Consider the following linear programming model.

Maximize Z $=50$ x$1+30$ x$2$

s$.$t$.$

$2$ x$1+3$ x$2<=15($Constraint $1)$

$-5$ x$1+2$ x$2<=3($Constraint $2)$

x$1<=4($Constraint $3)$

x$1-$ x$2<=3($Constraint $4)$

x$1,$ x$2>=0($Non$-$negativity constraints$)$

a$.$ Draw the graphical representation of the model indicating each constraint and the

objective function clearly. Make sure that you clearly label all the constraints and the

objective function.

b$.$ Give an upper bound on the number of corner point solutions of the problem by using the

formula.

c$.$ Identify the feasible region on the graph.

d$.$ Identify all corner point solutions indicating how you determine them verbally, showing

them on the graph and labeling them appropriately.

e$.$ Examining the corner point solutions on the graph, indicate whether each is feasible,

infeasible, or inconsistent.

f$.$ Find the optimal solution$($s$)$ to the given model by using graphical method. Explain your

steps in finding the optimal solution$($s$).$

g$.$ Draw the objective function contour on the graph where it is equal to optimal objective

value.