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

The graphical solution set for the system of inequations $x-2y \leq 2$,$5x+2y \geq 10$,$4x+5y \leq 20$,$x \geq 0$,$y \geq 0$ is given by

The vertices of the feasible region for the constraints $x+y \leq 4$,$x \leq 2$,$y \leq 1$,$x+y \geq 1$,$x, y \geq 0$ are

If $z = ax + by$ where $a, b > 0$ subject to the constraints $x \leq 2, y \leq 2, x + y \geq 3, x \geq 0, y \geq 0$ has a minimum value at $(2, 1)$ only,then...

$A$ factory manufactures two types of screws, $A$ and $B$. Each type of screw requires the use of two machines, an automatic and a hand-operated one. It takes $4 \, \text{minutes}$ on the automatic and $6 \, \text{minutes}$ on the hand-operated machine to manufacture a package of screws $A$, while it takes $6 \, \text{minutes}$ on the automatic and $3 \, \text{minutes}$ on the hand-operated machine to manufacture a package of screws $B$. Each machine is available for at most $4 \, \text{hours}$ on any day. The manufacturer can sell a package of screws $A$ at a profit of $Rs. \, 7$ and screws $B$ at a profit of $Rs. \, 10$. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.

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