A school invests up to 50 thousand pounds in two plans…

Step 1. The model. Maximise $Z = 60x + 90y$ pounds subject to $x + y \le 50$, $x \ge 10$, $y \le 30$ and $y \ge 0$.

Step 2. The corners. They are $(10, 0)$, $(50, 0)$, $(20, 30)$ and $(10, 30)$. The point $(20, 30)$ is where $x + y = 50$ meets $y = 30$.

Step 3. The returns. At $(10, 0)$, $Z = 600$. At $(50, 0)$, $Z = 3000$. At $(20, 30)$, $Z = 1200 + 2700 = 3900$. At $(10, 30)$, $Z = 600 + 2700 = 3300$. The largest is $3900$.

Step 4. Say it. Invest $20$ thousand pounds in plan A and $30$ thousand pounds in plan B. The return is $3900$ pounds a year.