The maximum value of $z=7x+8y$ subject to the constraints $x+y \leq 20, y \geq 5, x \leq 10, x \geq 0, y \geq 0$ is

$A$ dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contain at least $8$ $units$ of vitamin $A$ and $10$ $units$ of vitamin $C$. Food $'I'$ contains $2$ $units/kg$ of vitamin $A$ and $1$ $unit/kg$ of vitamin $C$. Food $'II'$ contains $1$ $unit/kg$ of vitamin $A$ and $2$ $units/kg$ of vitamin $C$. It costs Rs $50$ per $kg$ to purchase Food $'I'$ and Rs $70$ per $kg$ to purchase Food $'II'$. Formulate this problem as a linear programming problem to minimise the cost of such a mixture.

The cost function $Z$ is given by $Z = 4x + 6y$. It is to be minimized. The feasible region for this function $Z$ is the shaded region represented in the following figure. Then the minimum value of $Z$ is and occurs at the point:

The maximum value of the objective function $Z = 3x + 2y$ for the linear constraints $x + y \leq 7$,$2x + 3y \leq 16$,$x \geq 0$,$y \geq 0$ is

One kind of cake requires $200 \,g$ of flour and $25 \,g$ of fat,and another kind of cake requires $100 \,g$ of flour and $50 \,g$ of fat. Find the maximum number of cakes which can be made from $5 \,kg$ of flour and $1 \,kg$ of fat,assuming that there is no shortage of the other ingredients used in making the cakes.

$A$ scholarship amount is given by $z = 550x + 300y$ and is to be distributed among $x$ boys and $y$ girls. From the graph given below,the maximum amount of scholarship is . . . . . . .