The shaded area in the figure below is the solution set for a certain linear programming problem. The linear constraints are given by:

The objective function $z = 4x + 5y$ subject to the constraints $2x + y \geq 7$,$2x + 3y \leq 15$,$y \leq 3$,$x \geq 0$,and $y \geq 0$ has a minimum value at which point?

Minimize the objective function $Z = 3x + 2y$ subject to the constraints: $x + y \geq 8$,$x + y \leq 5$,$x \geq 0$,$y \geq 0$.

Reshma wishes to mix two types of food $P$ and $Q$ in such a way that the vitamin contents of the mixture contain at least $8$ $units$ of vitamin $A$ and $11$ $units$ of vitamin $B$. Food $P$ costs Rs $60/kg$ and Food $Q$ costs Rs $80/kg$. Food $P$ contains $3$ $units/kg$ of Vitamin $A$ and $5$ $units/kg$ of Vitamin $B$ while food $Q$ contains $4$ $units/kg$ of Vitamin $A$ and $2$ $units/kg$ of vitamin $B$. Determine the minimum cost of the mixture. Let the mixture contain $x$ kg of food $P$ and $y$ kg of food $Q$. Therefore,$x \geq 0$ and $y \geq 0$. The given information can be compiled in a table as follows:

The minimum value for the $LPP$ $Z = 6x + 2y$,subject to $2x + y \geq 16$,$x \geq 6$,$y \geq 1$ is

$A$ factory makes tennis rackets and cricket bats. $A$ tennis racket takes $1.5 \text{ hours}$ of machine time and $3 \text{ hours}$ of craftsman's time in its making,while a cricket bat takes $3 \text{ hours}$ of machine time and $1 \text{ hour}$ of craftsman's time. In a day,the factory has the availability of not more than $42 \text{ hours}$ of machine time and $24 \text{ hours}$ of craftsman's time. If the profit on a racket and on a bat is $Rs. 20$ and $Rs. 10$ respectively,find the maximum profit of the factory when it works at full capacity.