The objective function $z=x_1+x_2$,subject to $x_1+x_2 \leq 10, -2x_1+3x_2 \leq 15, x_1 \leq 6, x_1, x_2 \geq 0$,has a maximum value at:

The feasible region of an $LPP$ is shown in the figure. If $z=11x+7y$,then the maximum value of $z$ occurs at

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...

$A$ wholesale merchant wants to start a cereal business with $Rs \ 24000$. Wheat costs $Rs \ 400$ per quintal and rice costs $Rs \ 600$ per quintal. He has a storage capacity of $200$ quintals of cereal. He earns a profit of $Rs \ 25$ per quintal on wheat and $Rs \ 40$ per quintal on rice. If he stores $x$ quintals of rice and $y$ quintals of wheat,then for maximum profit,the objective function is:

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