Maximum value of $Z=100 x+70 y$ subject to $2 x \geq 4, y \leq 3, x+y \leq 8, x, y \geq 0$ is

$A$ diet is to contain at least $80$ units of vitamin $A$ and $100$ units of minerals. Two foods $F_{1}$ and $F_{2}$ are available. Food $F_{1}$ costs $Rs. 4$ per unit and food $F_{2}$ costs $Rs. 6$ per unit. One unit of food $F_{1}$ contains $3$ units of vitamin $A$ and $4$ units of minerals. One unit of food $F_{2}$ contains $6$ units of vitamin $A$ and $3$ units of minerals. Formulate this as a linear programming problem. Find the minimum cost for a diet that consists of a mixture of these two foods and also meets the minimal nutritional requirements.

The minimum value of $z = 2x + 4y$ subject to constraints $x + 2y \geq 10$,$3x + y \geq 10$,$x \geq 0$,$y \geq 0$ is $....$

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?

The maximum value of $Z=3x+5y$,subject to the constraints $x+4y \leq 24$,$y \leq 4$,$x \geq 0$,$y \geq 0$ 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?