The constraints of an LPP with two decision variables $x, y$ are given to be $y\le 3x, 3x+4y\ge 15$ and $2x+y\le 10$ where $x\ge 0, y\ge 0$. The objective functions $z$ of this LPP is maximum at the point $\left(3.5, 3\right)$. Determine $z$ if max $z=30$.

Maximize $Z=3x+5y$

Subject to

$x+2y\le 20$

$x+y\le 15$

$y\le 5$

$x,y\ge 0$

Maximize $z=x+y$

subject to $x+2y\le 10$

$x+y\ge 1$

$y\le 4$

where, $x, y\ge 0$.

Minimize $z=3x+2y$

subject to $2x+y\ge 14$

$2x+3y\ge 22$

$x+y\ge 5$

where $x, y\ge 0$.

A manufacturer has installed three machines to produce two items $A$ and $B$ (say). Machine $M_1$ and $M_2$ are capable of being operated for at most $10$ hours and $12$ hours per day. Machine $M_3$ must operate for at least $6$ hours a day. To produce a unit of item $A$, the manufacturer has to operate $M_1, M_2, M_3$ for $2$ hours, $1$ hour and $2$ hours respectively. Machines $M_1, M_2$ and $M_3$ are to be used for $1$ hour, $3$ hours and $2$ hours respectively to produce one unit of item $B$. The manufacturer makes a profit of Rs. $8$ on each unit of $A$ and a profit of Rs. $6$ on each unit of item $B$. It is assumed that he can sell out all the items he can produce. Formulate this problem as an LPP in which profit is made maximum.

A factory is engaged in manufacturing two products $A$ and $B$ which involves lathe work, grinding and assembling. The cutting, grinding and assembling times required for one unit on $A$ are $2, 1$ and $1$ hours respectively. Similarly $3, 1$ and $3$ hours for one unit of $B$. The profit on $A$ and $B$ are $\mathrm{Rs}.2, \mathrm{Rs}.3$ per unit respectively. Assuming that there are available $300$ hours of lathe time, $200$ hours of grinding time and $240$ hours of assembling time, the manufacturer wants to produce different type of items $A, B, C$ in such a way that the profit turns out to be maximum. Formulate the above as an LPP.