Problem $1($This problem was originally designed and solved by one of the course TAs, Amr Allam$)$A small manufacturer of concrete slabs produces two types of slabs. It wants to determine how many slabs of each type $($X$1$ of type $1$ and X$2$ of type $2)$ to produce over the coming period to maximize profit. Each type of slab passes through three manufacturing processes $($A$,$ B$,$ and C$).$ Information related to the required hours for manufacturing and the available man$-$hour for each process is as follows:Slab type $1$ needs $2$ man$-$hours of process A$,3$ hours of process B and $1$ hour of process C while slab type $2$ needs $5$ hours of process A$,1$ hours of process B and $1$ hour of process C$.$ The profits from manufacturing and selling the slabs are: $2$ $$/$slab type $1$ and $1$ $$/$slab type $2.$The total man$-$hour available for the processes are: $($A$,60$ hr$),($B$,44$ hr$)$ and $($C$,18$ hr$).$The market research team in the company dictates that at most $($no more than$)10$ standard slabs of X$2$ type could be produced over the coming period.$($i$)$ Tabulate the data;$($ii$)$ Develop a linear programming model that will enable the company to maximize its profit;$($iii$)$ Find the optimal solution using graphical method,$($iv$)$ Find the optimal solution if the profits were: $1$ $$/$slab type $1$ and $3$ $$/$slab type $2.$