$A$ factory makes tennis rackets and cricket bats. $A$ tennis racket takes $1.5 \text{ hours}$ of machine time and $3 \text{ hours}$ of craftsman's time in its making,while a cricket bat takes $3 \text{ hours}$ of machine time and $1 \text{ hour}$ of craftsman's time. In a day,the factory has the availability of not more than $42 \text{ hours}$ of machine time and $24 \text{ hours}$ of craftsman's time. If the profit on a racket and on a bat is $Rs. 20$ and $Rs. 10$ respectively,find the maximum profit of the factory when it works at full capacity.

The correct constraints for the given feasible region are:

The maximum value of $z = 5x + 3y$ subject to constraints $3x + 5y \leq 15, x \geq 0, y \geq 0$ is :

$A$ 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?

$A$ manufacturer of electronic circuits has a stock of $200$ resistors,$120$ transistors and $150$ capacitors and is required to produce two types of circuits $A$ and $B$. Type $A$ requires $20$ resistors,$10$ transistors and $10$ capacitors. Type $B$ requires $10$ resistors,$20$ transistors and $30$ capacitors. If the profit on type $A$ circuit is $Rs. 50$ and that on type $B$ circuit is $Rs. 60$,formulate this problem as a $LPP$ so that the manufacturer can maximize his profit.

The solution set of the constraints $x+2y \leq 2000$,$x+y \leq 1500$,$y \leq 600$ and $x \geq 0$ does not include the point