If the difference between the maximum and minimum values of the objective function $z = 7x - 8y$ subject to the constraints $x + y \leqslant 20$,$y \geqslant 5$,$x, y \geqslant 0$ is $5k + 200$,then the value of $k$ is

The vertices of the feasible region for the constraints $x+y \leq 4$,$x \leq 2$,$y \leq 1$,$x+y \geq 1$,$x, y \geq 0$ are

The maximum value of $z=9x+11y$ subject to $3x+2y \leq 12$,$2x+3y \leq 12$,$x \geq 0$,$y \geq 0$ is . . . . . . .

$A$ manufacturing company makes two models $A$ and $B$ of a product. Each piece of Model $A$ requires $9$ labour hours for fabricating and $1$ labour hour for finishing. Each piece of Model $B$ requires $12$ labour hours for fabricating and $3$ labour hours for finishing. For fabricating and finishing,the maximum labour hours available are $180$ and $30$ respectively. The company makes a profit of Rs $8000$ on each piece of model $A$ and Rs $12000$ on each piece of Model $B$. How many pieces of Model $A$ and Model $B$ should be manufactured per week to realise a maximum profit? What is the maximum profit per week?

The production of item $A$ is $x$ and the production of item $B$ is $y$. If the corner points of the bounded feasible region are $(1,0), (2,0), (0,2)$ and $(0,1)$,then the maximum profit $z = 2000x + 5000y$ is $\ldots \ldots$

$A$ firm has to transport $1200$ packages using large vans which can carry $200$ packages each and small vans which can take $80$ packages each. The cost for engaging each large van is $Rs. 400$ and each small van is $Rs. 200$. Not more than $Rs. 3000$ is to be spent on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as an $LPP$ given that the objective is to minimize cost.