Simunye Ltd is planning to expand its product range by producing two new products: product BP and product GG $.$ Product BP and product GG will be produced from different combinations of the same resources. Only products BP and GG use these resources. Details of the estimated selling price and cost per unit for each product are shown below:

$\backslash$table$[[,$Product BP$,$Product GG$,$Resources Available$],[$Selling price,R$115,$R$120,-],[$Material A $($R$10$ per kg$),$R$20,$R$10,900$ kg$],[$Material B $($R$6$ per kg$),$R$12,$R$24,1750$ kg$],[\backslash$table$[[$Labour $($R$14$ per hour$)]],$R$28,$R$21,1250$ hours$],[\backslash$table$[[$Variable overhead $($R$4$ per machine$],[$hour$)]],$R$14,$R$18,2400$ machine hours$]]$

The total fixed costs of producing product BP and GG will be R $70000$ per month. The management accountant at Simunye is in the process of producing a linear programming model to determine the optimal monthly production plan for the two new products. The objective of Simunye Ltd is to earn the maximum total profit possible per month from the production of products BP and GG $.$

NB: The optimal solution can be found using the iso$-$contribution line at point C of the feasible region, at the intersection of the 'Material $A\text{'}$ and 'Material $B\text{'}$ resource constraint lines.

Required:

$1\mathrm{.}1$ Use linear programming graph and simultaneous equations to determine the optimal production plan for the two products BP and GG$.$ DO NOT DRAW A GRAPH. ONLY SHOW ALL CALCULATIONS FOR BOTH THE GRAPH AND EQUATIONS.

$(19)$

ROUND OFF ALL CALCULATIONS TO $2$ DECIMAL PLACES.

$1\mathrm{.}2$ Explain the term 'shadow prices.

$(2)$

$1\mathrm{.}3$ List any $3$ assumptions of linear programming.