$($b$)$ A company makes three products $x,Y,Z$ out of three raw materials $A,B,C.$ The number

of units of raw materials required to produce one unit of the product is as given in the table

below

The unit of profit contribution of the products X$,$ Y$,$ and Z is K$40,$ K$25$ and K$50$

respectively. The number of units of raw materials available are $36,60,$ and $45$ respectively.

i$.$ Determine the product mix that will maximize the total profit by using the simplex

algorithm.

ii$.$ From the final simplex table, write the solution to the dual and give economic

interpretation