$($a$)$ Consider the following system of constraints, associated with a linear programming problem:

$3x+y24$

$2x+2y20$

$x+3y24$

$x0$

$y0.$

Maximize $z=5x+3y.$

$($b$)$ An electronics firm manufactures two types of personal computers, a desktop model and a portable model. The production of a desktop computer requires a capital expenditure of R$4000$ and $40$ hours of labour. The production of a portable computer requires a capital expenditure of R$2500$ and $30$ hours of labour. The firm has R$200000$ capital and $2160$ labour hours available for production of desktop and portable computers.

i$.$ What is the maximum number of computers the company is capable of producing?$(5)$

ii$.$ If each desktop computer contributes a profit of R$3200$ and each portable contributes a profit of R$2200,$ how much profit will the company make by producing the maximum number of computers determined in $($i$),$ above?