Consider the following LP problem.

Maximize $,z=20x_A+50x_B$

Subject to

$0\mathrm{.}2x_A-0\mathrm{.}8x_{B}0$

$x_{A}100$

$2x_A+4x_{B}240$

$10x_A+15x_{B}1500$

$x_A,x_{B}0$

$[$III$]$

$($a$)$ On a two$-$dimensional graph, show the feasible region and its extreme points.

$($b$)$ Which of the constraints $($excluding the non$-$negativity ones$)$ is$($are$)$ redundant? $($IV$)$

$($c$)$ For each extreme point, determine the optimal objective function value and the optimal values of the decision variables $x_1$ and $x_2.$

$($d$)$ In the optimal solution of the problem, which of the constraints $($excluding the nonnegativity ones$)$ are binding?

$($e$)$ Suppose that the objective was changed to minimization and a new constraint $10x_A+25x_{B}500$ was added to the problem. What effect would this have on the feasible region and the optimal solution?