The maximum value of $z=4x+2y$,subject to the constraints $3x+4y \geqslant 12$,$x+y \leqslant 5$,$x, y \geqslant 0$ is

$A$ 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 $Z = 5x + 2y$,subject to the constraints $2x - y \geq 2$,$x + 2y \leq 8$,and $x, y \geq 0$,is:

The maximum value of the objective function $z=4x+6y$ subject to the constraints $3x+2y \leq 12$,$x+y \geq 4$,$x, y \geq 0$ is

The minimum value of $z = 10x + 25y$ subject to the constraints $0 \leq x \leq 3$,$0 \leq y \leq 3$,and $x + y \geq 5$ is:

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: