Corner points of the feasible region for an $LPP$ are $(0,2), (3,0), (6,0), (6,8)$ and $(0,5)$. Let $Z = 4x + 6y$ be the objective function. The minimum value of $Z$ occurs at

For a linear programming problem,the objective function is $Z = 10500x + 9000y$. If the corner points of the bounded feasible region are $(0,0)$,$(40,0)$,$(30,20)$,and $(0,50)$,then the maximum value of $Z$ is . . . . . . .

An aeroplane can carry a maximum of $200$ passengers. $A$ profit of $Rs. 1000$ is made on each executive class ticket and a profit of $Rs. 600$ is made on each economy class ticket. The airline reserves at least $20$ seats for executive class. However,at least $4$ times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit?

For the objective function $Z = 4x + y$ subject to the constraints $x + y \leq 50$,$3x + y \leq 90$,$x \geq 0$,$y \geq 0$,whose corner points of the feasible region are $(0,0)$,$(30,0)$,$(20,30)$,and $(0,50)$,the maximum value of $Z$ is . . . . . . .

The feasible solution for a $LPP$ is shown in the figure. Let $z=3x-4y$ be the objective function. The minimum value of $z$ occurs at:

The corner points of the feasible region determined by the system of linear constraints are $(2, 72)$,$(15, 20)$,and $(40, 15)$. Let $Z = 6x + 3y$ be the objective function. The minimum value of $Z$ occurs at: