The maximum value of $z = 4x + y$ subject to the constraints $x + y \leq 5$, $2x + y \leq 7$, $3x + 2y \leq 11$, $x \geq 0$, $y \geq 0$ is

Solution of the following $LP$ problem
Minimize $z = -3x + 2y$
subject to $0 \leq x \leq 4, 1 \leq y \leq 6, x + y \leq 5$ is $.....$

The maximum value of the objective function $z=4x+5y$ subject to the constraints $2x+3y \leq 12$,$2x+y \leq 8$ and $x \geq 0, y \geq 0$ is:

$A$ fruit grower can use two types of fertilizer in his garden,brand $P$ and brand $Q$. The amounts (in $kg$) of nitrogen,phosphoric acid,potash,and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least $240 \, kg$ of phosphoric acid,at least $270 \, kg$ of potash,and at most $310 \, kg$ of chlorine.
If the grower wants to maximize the amount of nitrogen added to the garden,how many bags of each brand should be added? What is the maximum amount of nitrogen added?

	Brand $P$ ($kg$ per bag)	Brand $Q$ ($kg$ per bag)
Nitrogen	$3$	$3.5$
Phosphoric acid	$1$	$2$
Potash	$3$	$1.5$
Chlorine	$1.5$	$2$

The maximum value of $Z=10 x+25 y$ subject to $0 \leq x \leq 3, 0 \leq y \leq 3, x+y \leq 5, x \geq 0, y \geq 0$ is

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?