Find the point at which the objective function $Z = x + y$ attains its maximum value subject to the constraints $x + 2y \leq 70$, $2x + y \leq 95$, $x \geq 0$, and $y \geq 0$.

$A$ 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.

The shaded part of the given figure indicates the feasible region. Then the constraints are

The difference between the maximum and minimum values of the objective function $Z = 3x + 5y$, subject to the constraints $x + 3y \leq 60$, $x + y \geq 10$, $x - y \leq 0$, and $x, y \geq 0$, is

In order to supplement daily diet,a person wishes to take some $X$ and some $Y$ tablets. The contents of iron,calcium and vitamins in $X$ and $Y$ (in milligrams per tablet) are given as below:

Tablets	Iron	Calcium	Vitamin
$X$	$6$	$3$	$2$
$Y$	$2$	$3$	$4$

The person needs at least $18$ milligrams of iron,$21$ milligrams of calcium and $16$ milligrams of vitamins. The price of each tablet of $X$ and $Y$ is $Rs. 2$ and $Rs. 1$ respectively. How many tablets of each should the person take in order to satisfy the above requirement at the minimum cost?

The objective function $z=x_1+x_2$,subject to $x_1+x_2 \leq 10, -2x_1+3x_2 \leq 15, x_1 \leq 6, x_1, x_2 \geq 0$,has a maximum value at: