The maximum value of $Z=10 x+25 y$ subject to $0 \leq x \leq 3, 0 \leq y \leq 3, x+y \leq 5, x \geq 0, y \geq 0$ 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:

$A$ 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?

The shaded region in the following figure is the solution set of the inequations:

$A$ 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 maximum value of $z=3x+4y$,subject to the constraints $x+y \leq 40$,$x+2y \leq 60$ and $x, y \geq 0$ is