The maximum value of $z = 3x + 5y$ subject to the constraints $3x + 2y \leq 18$,$x \leq 4$,$y \leq 6$,$x, y \geq 0$,is

The shaded area in the figure below is the solution set for a certain linear programming problem. The linear constraints are given by:

The maximum value of $z = 9x + 13y$ subject to the constraints $2x + 3y \leq 18$,$2x + y \leq 10$,$x \geq 0$,$y \geq 0$ is:

$A$ firm has to transport $1200$ packages using large vans which can carry $200$ packages each and small vans which can take $80$ packages each. The cost for engaging each large van is $Rs. 400$ and each small van is $Rs. 200$. Not more than $Rs. 3000$ is to be spent on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as an $LPP$ given that the objective is to minimize cost.

Find the maximum value of $z = 2x + 6y$ subject to the constraints $-x + y \leq 1$,$2x + y \leq 2$,$x \geq 0$,and $y \geq 0$.

$A$ 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	$1.5$	$2$