A company produces two products, A and $B.$ The sales volume for A has to be at least $80\%$

of the total sales of both A and B$.$ However, the company cannot sell more than $100$ units of

A per day. Both products use one raw material, of which the maximum daily availability is

$240lb.$ The usage rates of the raw material are $2lb$ per unit of A and $4lb$ per unit of $B.$ The

profit units for A and B are $$20$ and $$50,$ respectively.

a$.$ Formulate an LP to maximize profit.

b$.$ Graphically solve the LP$,$ clearly present the optimal solution and optimal value

c$.$ Determine the range of profit earned by selling one A that keeps the current solution

optimal. $($Use graphical solution$)$

d$.$ Determine the shadow price of the raw material using graphical solution.