The minimum value of $Z=150 x+200 y$ for the given constraints $\begin{aligned} & 3 x+5 y \geq 30 \\ & x+y \geq 8 ; x \geq 0, y \geq 0 \text { is } \end{aligned}$

Corner points of the feasible region are found by intersecting the boundary lines.

Intersection of $3x + 5y = 30$ and $x + y = 8$:
Subtracting $3(x+y=8)$ from first: $2y = 6 \Rightarrow y = 3,\ x = 5$. Point: $(5, 3)$

Other corner points: $(10, 0)$ and $(0, 8)$.

Evaluating $Z = 150x + 200y$:
At $(10, 0)$: $Z = 1500$
At $(5, 3)$: $Z = 750 + 600 = 1350$
At $(0, 8)$: $Z = 1600$

Since the feasible region is unbounded, verify that no point in the region gives $Z < 1350$. Testing $(6,4)$: $Z = 900 + 800 = 1700 > 1350$. The half-plane $150x + 200y < 1350$ does not intersect the feasible region.

Minimum value of $Z = 1350$ at $(5, 3)$.