Cake-$A$ requires $200\, g$ of flour and $25\, g$ of fat. Cake-$B$ 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. The mathematical form of this $LPP$ is $.....$

If $Z=10x+25y$ subject to $0 \leq x \leq 3, 0 \leq y \leq 3, x+y \leq 5, x \geq 0, y \geq 0$,then $Z$ is maximum at the point:

$A$ manufacturer produces two models of bikes: Model $X$ and Model $Y$. Model $X$ takes $6$ man-hours to make per unit,while Model $Y$ takes $10$ man-hours per unit. There is a total of $450$ man-hours available per week. Handling and marketing costs are $Rs. 2000$ and $Rs. 1000$ per unit for Models $X$ and $Y$ respectively. The total funds available for these purposes are $Rs. 80,000$ per week. Profits per unit for Models $X$ and $Y$ are $Rs. 1000$ and $Rs. 500$ respectively. How many bikes of each model should the manufacturer produce so as to yield a maximum profit? Find the maximum profit.

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

The graphical solution set for the system of inequations $x-2y \leq 2$,$5x+2y \geq 10$,$4x+5y \leq 20$,$x \geq 0$,$y \geq 0$ is given by