1. **State the problem:** A farmer has 1000 acres of land to plant wheat and barley. Wheat requires 2 hours of labor per acre, barley requires 1 hour per acre, and the farmer has 1200 labor hours available. Wheat yields 200 profit per acre, barley 150 profit per acre. The farmer must plant at least 200 acres of wheat. We want to find how many acres of each crop to plant to maximize profit. 2. **Define variables:** Let $x$ = acres of wheat planted. Let $y$ = acres of barley planted. 3. **Write constraints:** - Land constraint: $$x + y \leq 1000$$ - Labor constraint: $$2x + y \leq 1200$$ - Wheat minimum: $$x \geq 200$$ - Non-negativity: $$x \geq 0, y \geq 0$$ 4. **Profit function to maximize:** $$P = 200x + 150y$$ 5. **Analyze constraints and find feasible region corners:** - From land: $y \leq 1000 - x$ - From labor: $y \leq 1200 - 2x$ - Wheat minimum: $x \geq 200$ 6. **Find intersection points of constraints:** - Intersection of $x + y = 1000$ and $2x + y = 1200$: $$\begin{cases} x + y = 1000 \\ 2x + y = 1200 \end{cases}$$ Subtract first from second: $$2x + y - (x + y) = 1200 - 1000 \Rightarrow x = 200$$ Plug back: $$200 + y = 1000 \Rightarrow y = 800$$ So point A: $(200, 800)$ - Check point B: wheat minimum and land constraint: At $x=200$, $y \leq 800$ (from land), so point B is $(200, 800)$ same as A. - Check point C: wheat minimum and labor constraint: At $x=200$, $y \leq 1200 - 2(200) = 1200 - 400 = 800$, so point C is also $(200, 800)$. - Check point D: wheat minimum and $y=0$: At $x=200$, $y=0$ is feasible. - Check point E: land constraint and $y=0$: $$x + 0 \leq 1000 \Rightarrow x \leq 1000$$ But $x \geq 200$, so $x=1000$ is feasible. - Check labor constraint at $y=0$: $$2x \leq 1200 \Rightarrow x \leq 600$$ So max $x$ at $y=0$ is 600. 7. **Evaluate profit at corner points:** - At $(200, 800)$: $$P = 200(200) + 150(800) = 40000 + 120000 = 160000$$ - At $(600, 0)$: $$P = 200(600) + 150(0) = 120000 + 0 = 120000$$ - At $(200, 0)$: $$P = 200(200) + 150(0) = 40000 + 0 = 40000$$ 8. **Conclusion:** Maximum profit is at $(200, 800)$ acres. **Answer:** Plant 200 acres of wheat and 800 acres of barley to maximize profit.