The difference between the maximum and minimum values of the objective function $Z = 3x + 5y$, subject to the constraints $x + 3y \leq 60$, $x + y \geq 10$, $x - y \leq 0$, and $x, y \geq 0$ is

$A$ manufacturing company makes two models $A$ and $B$ of a product. Each piece of Model $A$ requires $9$ labour hours for fabricating and $1$ labour hour for finishing. Each piece of Model $B$ requires $12$ labour hours for fabricating and $3$ labour hours for finishing. For fabricating and finishing,the maximum labour hours available are $180$ and $30$ respectively. The company makes a profit of Rs $8000$ on each piece of model $A$ and Rs $12000$ on each piece of Model $B$. How many pieces of Model $A$ and Model $B$ should be manufactured per week to realise a maximum profit? What is the maximum profit per week?

The objective function $Z = 4 x_1 + 5 x_2$,subject to $2 x_1 + x_2 \geq 7$,$2 x_1 + 3 x_2 \leq 15$,$x_2 \leq 3$,$x_1, x_2 \geq 0$ has minimum value at the point

There are two types of fertilisers $F_{1}$ and $F_{2}$. $F_{1}$ consists of $10\%$ nitrogen and $6\%$ phosphoric acid,and $F_{2}$ consists of $5\%$ nitrogen and $10\%$ phosphoric acid. After testing the soil conditions,a farmer finds that she needs at least $14\,kg$ of nitrogen and $14\,kg$ of phosphoric acid for her crop. If $F_{1}$ costs $Rs\,6/kg$ and $F_{2}$ costs $Rs\,5/kg$,determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

Solution of the following $LP$ problem
Minimize $z = -3x + 2y$
subject to $0 \leq x \leq 4, 1 \leq y \leq 6, x + y \leq 5$ is $.....$

$A$ 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. What number of rackets and bats must be made if the factory is to work at full capacity?