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

$A$ dietician wishes to mix together two kinds of food $X$ and $Y$ in such a way that the mixture contains at least $10$ units of vitamin $A$,$12$ units of vitamin $B$,and $8$ units of vitamin $C$. The vitamin contents of one $kg$ of food are given below:

Food	Vitamin $A$	Vitamin $B$	Vitamin $C$
$X$	$1$	$2$	$3$
$Y$	$2$	$2$	$1$

One $kg$ of food $X$ costs $Rs. 16$ and one $kg$ of food $Y$ costs $Rs. 20$. Find the least cost of the mixture which will produce the required diet?

The maximum value of $Z = 4x + 5y$, subject to the constraints $3x + y \leq 15$, $3x + 4y \leq 24$, $x \geq 0$, $y \geq 0$ is

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

$A$ wholesale dealer wants to start a business with $Rs. 2,40,000$. The cost price of a quintal of wheat is $Rs. 2000$ and the cost price of a quintal of rice is $Rs. 3000$. He has space capacity for $200$ quintals of grain. The profit from the sale of one quintal of wheat is $Rs. 125$ and that from one quintal of rice is $Rs. 200$. If he has $x$ quintals of rice and $y$ quintals of wheat,then the objective function for the maximum profit is $....$

For the following shaded region,the linear constraints are: