A farmer needs at least $50$ cows for a new herd. He can buy brown cows $($x$)$ at N$ $6000$ each and black cows $($y$)$ at N$ $4000$ each and he can spend a total of no more than N$ $240000.$ He wants at least $30$ brown cows. On selling the cows he makes a profit of N$ $1000\mathrm{????}\mathrm{????}$ each brown cow and N$ $1200$ on each black cow.

a$)$ Write all the inequalities.

b$)$ Draw a sketch graph and shade the feasible region.

c$)$ Find the coordinates of the corner points of this region.

d$)$ What combination will ensure the biggest profit? 

e$)$ What is this profit?