The difference between the maximum value and minimum value of the objective function $z = 3x + 5y$ subject to the constraints $x + 3y \leqslant 60$,$x + y \geqslant 10$,$x - y = 0$,and $x, y \geqslant 0$ is:

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:

If the feasible region is as shown in the figure,then the related inequalities are:

$A$ manufacturing company produces two items,$A$ and $B$. Each item must be processed by two machines,$I$ and $II$. Machine $I$ can be operated for a maximum of $10$ hours $40$ minutes ($640$ minutes). It takes $20$ minutes for an item $A$ and $15$ minutes for an item $B$. Machine $II$ can be operated for a maximum of $8$ hours $20$ minutes ($500$ minutes). It takes $5$ minutes for an item $A$ and $8$ minutes for an item $B$. The profit per item of $A$ is ₹ $25$ and per item of $B$ is ₹ $18$. The formulation of an $L.P.P.$ to maximize the profit (where $x$ is the number of items $A$ and $y$ is the number of items $B$) 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?

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?