The feasible region for the constraints $x-2 \leqslant y$,$x \geqslant y-1$,$x \geqslant 2$,$y \leqslant 4$,$x, y \geqslant 0$ is represented by:

$A$ farmer mixes two brands $P$ and $Q$ of cattle feed. Brand $P$,costing $Rs. 250$ per bag,contains $3$ units of nutritional element $A$,$2.5$ units of element $B$ and $2$ units of element $C$. Brand $Q$ costing $Rs. 200$ per bag contains $1.5$ units of nutritional element $A$,$11.25$ units of element $B$,and $3$ units of element $C$. The minimum requirements of nutrients $A$,$B$ and $C$ are $18$ units,$45$ units and $24$ units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

There are two types of fertilisers $F_{1}$ and $F_{2}$. $F_{1}$ consists of $10\%$ nitrogen and $6\%$ phosphoric acid,and $F_{2}$ consists of $5\%$ nitrogen and $10\%$ phosphoric acid. After testing the soil conditions,a farmer finds that she needs at least $14\,kg$ of nitrogen and $14\,kg$ of phosphoric acid for her crop. If $F_{1}$ costs $Rs\,6/kg$ and $F_{2}$ costs $Rs\,5/kg$,determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

The function to be maximized is given by $Z=2x+y$. The feasible region for this function $Z$ is the shaded region shown in the figure. The maximum value of $Z$ is . . . . . . and occurs at the point . . . . . . .

The shaded area in the given figure is a solution set for some system of inequalities. The maximum value of the function $z=4x+3y$ subject to linear constraints given by the system is

The minimum value of $Z = 3x + y$, subject to the constraints $2x + 3y \leq 6$, $x + y \geq 1$, $x \geq 0$, $y \geq 0$ is....