The maximum value of $$P=500 x+400 y$$ for the given constraints $$x+y \leq 200, \quad x \geq 20, \quad y \geq 4 x, \quad y \geq 0$$ is

$$ \text { The maximum value of } Z=3 x+4 y \text { for the given constraints } x+2 y \leq 76,2 x+y \leq 104, x \geq 0, y \geq 0 \text { is } $$

The minimum value of $$Z=150 x+200 y$$ for the given constraints

$$\begin{aligned} & 3 x+5 y \geq 30 \\ & x+y \geq 8 ; x \geq 0, y \geq 0 \text { is } \end{aligned}$$

The feasible region for the inequations $$x+2 y \geq 4,2 x+y \leq 6, x, y \geq 0$$ is