In the following figure, the shaded region represents the system of constraints:

$A$ farmer mixes two brands $P$ and $Q$ of cattle feed. Brand $P$,costing $Rs. 250$ per bag,contains $3$ units of nutritional element $A$,$2.5$ units of element $B$ and $2$ units of element $C$. Brand $Q$ costing $Rs. 200$ per bag contains $1.5$ units of nutritional element $A$,$11.25$ units of element $B$,and $3$ units of element $C$. The minimum requirements of nutrients $A$,$B$ and $C$ are $18$ units,$45$ units and $24$ units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

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?

$A$ diet of a sick person must contain at least $4000$ units of vitamins,$50$ units of proteins,and $1400$ calories. Two foods $A$ and $B$ are available at a cost of ₹ $4$ and ₹ $3$ per unit respectively. If one unit of $A$ contains $200$ units of vitamins,$1$ unit of protein,and $40$ calories,while one unit of food $B$ contains $100$ units of vitamins,$2$ units of protein,and $40$ calories,formulate the problem so that the diet is the cheapest.

$A$ 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.

$A$ manufacturer produces nuts and bolts. It takes $1\, hour$ of work on machine $A$ and $3\, hours$ on machine $B$ to produce a package of nuts. It takes $3\, hours$ on machine $A$ and $1\, hour$ on machine $B$ to produce a package of bolts. He earns a profit of $Rs.\,17.50$ per package on nuts and $Rs.\,7$ per package on bolts. How many packages of each should be produced each day so as to maximise his profit,if he operates his machines for at the most $12\, hours$ a day?