State advantages of linear programming problems.

Maximize $\mathrm{Z}=3\mathrm{x}+2\mathrm{y}$ subject to $x+2y\le 10,3x+y\le 15,x,y\ge 0$

A manufacturer produces nuts and bolts for industrial machinery. It takes $1$ hour of work on machine $A$ and $3$ hours on machine $B$ to produces a packet of nuts while it takes $3$ hours on machine $A$ and $1$ hour on machine $B$ to produce a packet of bolts. He earns a profit $₹17.50$ per packet on nuts and $₹7$per packet on bolts. How many packets of each should be produced each day so as to maximise his profit if he operates each machine for at the most $12$ hours a day? Also find the maximum profit.

A manufacturer produces two types of soap bars using two machines, $A$ and $B$. $A$ is operated for $2$ minutes and $B$ for $3$ minutes to manufacture the first type, while it takes $3$ minutes on machine $A$ and $5$ minutes on machine $B$ to manufacture the second type. Each machine can be operated at the most for $8$ hours per day. The two types of soap bars are sold at a profit of $₹0.25$ and $₹0.50$ each. Assuming that the manufacturer can sell all the soap bars he can manufacture, how many bars of soap of each type should be manufactured per day so as to maximize his profit?

State advantages of linear programming problems.

The corner points of the feasible region determined by the following system of linear inequalities:
$2x+y\le 10,x+3y\le 15,x,y\ge 0$ are $\left(0,0\right),\left(5,0\right),\left(3,4\right)$ and $\left(0,5\right)$
Let $Z=px+qy,$ where $p,q>0.$
Condition on $p$ and $q$ so that the maximum of $Z$ occurs at both $\left(3,4\right)$ and $\left(0,5\right)$ is

A dealer wishes to purchase a number of fans and sewing machines. He has only $₹5760$ to invest and space for at most $20$ items. A fan costs him $₹360$ and a sewing machine,$₹240$. He expects to gain $₹22$ on a fan and $₹18$ on a sewing machine. Assuming that he can sell all the items he can buy, how should he invest the money in order to maximise the profit?

A carpenter has $90, 80$ and $50$ running feet respectively of teak wood, plywood and rosewood which is used to produce product $A$ and product $B$. Each unit of product $A$ requires $2,1$ and $1$ running feet and each unit of product $B$ requires $1, 2$ and $1$ running feet of teak wood, plywood and rosewood respectively. If product $A$ is sold for $₹48$ per unit and product $B$ is sold for $₹40$ per unit, how many units of product $A$ and product $B$ should be produced and sold by the carpenter, in order to obtain the maximum gross income?

Formulate the above as a Linear Programming Program and solve it, indicating clearly the feasible region in the graph.

Maximize $z=3x-8y$

subject to $4x+4y\le 4$

$4x+8y\ge 16$

where, $x, y\ge 0$.