The minimum value of $z = 3x + 5y$, subject to constraints $x \leq 80$, $y \geq 60$, $x + y \leq 200$, $x, y \geq 0$ occurs at the point...

For the following shaded region,the linear constraints are:

The maximum value of $z=6x+8y$ subject to the constraints $x-y \geq 0$,$x+3y \leq 12$,$x \geq 0$,$y \geq 0$ is:

The graphical solution set of the system of in-equations $x+y \leq 70, x+2y \leq 100, 2x+y \leq 120, x \geq 0, y \geq 0$ is given by:

$A$ manufacturer makes two types of toys $A$ and $B$. Three machines are needed for this purpose and the time (in $minutes$) required for each toy on the machines is given below:

Types of Toys	Machine-$I$	Machine-$II$	Machine-$III$
$A$	$12$	$18$	$6$
$B$	$6$	$0$	$9$

Each machine is available for a maximum of $6 \, hours$ $(360 \, minutes)$ per day. If the profit on each toy of type $A$ is $Rs. \, 7.50$ and that on each toy of type $B$ is $Rs. \, 5$,find the number of toys of each type that should be manufactured in a day to get maximum profit.

If the difference between the maximum and minimum values of the objective function $z = 7x - 8y$ subject to the constraints $x + y \leqslant 20$,$y \geqslant 5$,$x, y \geqslant 0$ is $5k + 200$,then the value of $k$ is