The feasible region represented by the given constraints $2x + 3y \geqslant 12$,$-x + y \leqslant 3$,$x \leqslant 4$,$y \geqslant 3$ is denoted by

The minimum value of $Z = 5x + 8y$ subject to the constraints $x + y \geq 5$,$0 \leq x \leq 4$,$y \geq 2$ is:

$A$ cooperative society of farmers has $50$ hectares of land to grow two crops $X$ and $Y$. The profit from crops $X$ and $Y$ per hectare are estimated as Rs $10,500$ and Rs $9,000$ respectively. To control weeds,a liquid herbicide has to be used for crops $X$ and $Y$ at rates of $20$ litres and $10$ litres per hectare. Further,no more than $800$ litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society?

An airplane can carry a maximum of $250$ passengers. $A$ profit of $Rs \ 1500$ is made on each executive class ticket and a profit of $Rs \ 900$ is made on each economy class ticket. The airline reserves at least $30$ seats for executive class. However, at least $4$ times as many passengers prefer to travel by economy class than by executive class. Let $x_1$ be the number of passengers in executive class and $x_2$ be the number of passengers in economy class. Formulate the Linear Programming Problem $(LPP)$ to maximize the profit for the airline.

In the following figure, the shaded region represents the system of constraints:

For the following shaded region,the linear constraints are: