The graphical solution set of the system of in-equations $x+y \leq 70, x+2y \leq 100, 2x+y \leq 120, x \geq 0, y \geq 0$ is given by:

$A$ man rides his motorcycle at the speed of $50 \, km/h$. He has to spend $Rs. \, 2$ per $km$ on petrol. If he rides it at a faster speed of $80 \, km/h$,the petrol cost increases to $Rs. \, 3$ per $km$. He has at most $Rs. \, 120$ to spend on petrol and one hour's time. He wishes to find the maximum distance that he can travel. Express this problem as a linear programming problem.

In the following figure, the shaded region represents the system of constraints:

The maximum value of $z=9x+11y$ subject to $3x+2y \leq 12$,$2x+3y \leq 12$,$x \geq 0$,$y \geq 0$ 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:

$A$ dietician has to develop a special diet using two foods $X$ and $Y$. Each packet (containing $30 \ g$) of food $X$ contains $12$ units of calcium,$4$ units of iron,$6$ units of cholesterol and $6$ units of vitamin $A$. Each packet of the same quantity of food $Y$ contains $3$ units of calcium,$20$ units of iron,$4$ units of cholesterol and $3$ units of vitamin $A$. The diet requires at least $240$ units of calcium,at least $460$ units of iron and at most $300$ units of cholesterol. The corner points of the feasible region are: