The coordinates of the corner points of the bounded feasible region are $(0, 10)$,$(5, 5)$,$(15, 15)$,and $(0, 20)$. The minimum value of the objective function $z = 3x + 9y$ is . . . . . . .

The feasible region (shaded) for a $LPP$ is shown in the adjacent figure. Maximize $Z = 5x + 7y$.

The solution set of the constraints $x + 2y \geq 11$,$3x + 4y \leq 30$,$2x + 5y \leq 30$,$x \geq 0$,$y \geq 0$ includes the point:

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:

For a linear programming problem,the objective function is $Z = 3x + 2y$. If the corner points of the bounded feasible region are $(12, 0)$,$(4, 2)$,$(1, 5)$,and $(1, 10)$,then the maximum value of $Z$ is . . . . . . .