For the following shaded region,the linear constraints are:

$A$ 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.

For the linear programming problem, $x + 2y \leq 10$, $3x + y \leq 12$, $x, y \geq 0$, the maximum value of $z = 5x + 10y$ occurs at every point on the line segment joining the points..

The minimum value for the $LPP$ $Z = 6x + 2y$,subject to $2x + y \geq 16$,$x \geq 6$,$y \geq 1$ is

There are two factories located at place $P$ and place $Q$. From these locations,a certain commodity is to be delivered to each of the three depots situated at $A, B$ and $C$. The weekly requirements of the depots are $5, 5$ and $4$ units respectively,while the production capacities of the factories at $P$ and $Q$ are $8$ and $6$ units respectively. The cost of transportation per unit is given below:

From/To	$A$	$B$	$C$
$P$	$160$	$100$	$150$
$Q$	$100$	$120$	$100$

How many units should be transported from each factory to each depot in order that the transportation cost is minimum? What will be the minimum transportation cost?

Cake-$A$ requires $200\, g$ of flour and $25\, g$ of fat. Cake-$B$ requires $100\, g$ of flour and $50\, g$ of fat. Find the maximum number of cakes which can be made from $5\, kg$ of flour and $1\, kg$ of fat. The mathematical form of this $LPP$ is $.....$