The shaded region in the following figure is the solution set of the inequations:

$A$ 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?

$A$ manufacturer of electronic circuits has a stock of $200$ resistors,$120$ transistors and $150$ capacitors and is required to produce two types of circuits $A$ and $B$. Type $A$ requires $20$ resistors,$10$ transistors and $10$ capacitors. Type $B$ requires $10$ resistors,$20$ transistors and $30$ capacitors. If the profit on type $A$ circuit is $Rs. 50$ and that on type $B$ circuit is $Rs. 60$,formulate this problem as a $LPP$ so that the manufacturer can maximize his profit.

$A$ company manufactures two types of sweaters: type $A$ and type $B.$ It costs $Rs. 360$ to make a type $A$ sweater and $Rs. 120$ to make a type $B$ sweater. The company can make at most $300$ sweaters and spend at most $Rs. 72000$ a day. The number of sweaters of type $B$ cannot exceed the number of sweaters of type $A$ by more than $100.$ The company makes a profit of $Rs. 200$ for each sweater of type $A$ and $Rs. 120$ for every sweater of type $B.$ Formulate this problem as a $LPP$ to maximize the profit to the company.

The minimum value of the objective function $Z = 5x + 8y$,subject to the constraints $x + y \geq 5$,$x \leq 4$,$y \leq 2$,$x \geq 0$,and $y \geq 0$,occurs at the point:

The minimum value of $z = 10x + 25y$ subject to the constraints $0 \leq x \leq 3$,$0 \leq y \leq 3$,and $x + y \geq 5$ is: