The maximum value of $z=4x+2y$,subject to the constraints $3x+4y \geqslant 12$,$x+y \leqslant 5$,$x, y \geqslant 0$ is

$A$ 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.

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

$A$ manufacturer has three machines $I, II$ and $III$ installed in his factory. Machines $I$ and $II$ are capable of being operated for at most $12 \, hours$ whereas machine $III$ must be operated for at least $5 \, hours$ a day. She produces only two items $M$ and $N$ each requiring the use of all the three machines. The number of hours required for producing $1$ unit of each of $M$ and $N$ on the three machines are given in the following table:

Items	Machine $I$	Machine $II$	Machine $III$
$M$	$1$	$2$	$1$
$N$	$2$	$1$	$1.25$

She makes a profit of $Rs. \, 600$ and $Rs. \, 400$ on items $M$ and $N$ respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?

The $L$.$P$.$P$. to maximize $z=x+y$,subject to $x+y \leq 30, x \leq 15, y \leq 20, x+y \geq 15$,and $x, y \geq 0$ has