The shaded area in the given figure is a solution set for some system of inequations. The maximum value of the function $z=10x+25y$ subject to the linear constraints given by the system is

The maximum value of $Z=3x+5y$,subject to the constraints $3x+2y \leq 18$,$x \leq 4$,$y \leq 6$,and $x, y \geq 0$ is

The feasible region for the constraints $x-y \geqslant 0$,$x-5y \leqslant -5$,$x \geqslant 0$,$y \geqslant 0$ is shown by the figure:

The minimum value of the objective function $z = 4x + 6y$ subject to the constraints $x + 2y \geq 80$,$3x + y \geq 75$,and $x, y \geq 0$ 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.

$A$ production unit makes a special type of metal chip by combining copper and brass. The standard weight of the chip must be at least $5 \text{ gms}$. The basic ingredients,i.e.,copper and brass,cost $₹8$ and $₹5$ per $\text{gm}$ respectively. Durability considerations dictate that the metal chip must not contain more than $4 \text{ gms}$ of brass and should contain a minimum of $2 \text{ gms}$ of copper. Then,the minimum cost of the metal chip satisfying the above conditions is: