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

$A$ 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?

If $Z=10x+25y$ subject to $0 \leq x \leq 3, 0 \leq y \leq 3, x+y \leq 5, x \geq 0, y \geq 0$,then $Z$ is maximum at the point:

$A$ company makes $3$ models of calculators: $A, B$ and $C$ at factory $I$ and factory $II.$ The company has orders for at least $6400$ calculators of model $A, 4000$ calculators of model $B$ and $4800$ calculators of model $C.$ At factory $I, 50$ calculators of model $A, 50$ of model $B$ and $30$ of model $C$ are made every day; at factory $II, 40$ calculators of model $A, 20$ of model $B$ and $40$ of model $C$ are made every day. It costs $Rs. 12000$ and $Rs. 15000$ each day to operate factory $I$ and $II,$ respectively. Find the number of days each factory should operate to minimize the operating costs and still meet the demand.

The minimum value of $z = 3x + 5y$, subject to constraints $x \leq 80$, $y \geq 60$, $x + y \leq 200$, $x, y \geq 0$ occurs at the point...

For the following shaded region,the linear constraints are: