1. **State the problem:** You have a test with geography problems worth 5 points each and history problems worth 15 points each. You can solve a geography problem in 1 minute and a history problem in 5 minutes. You have 60 minutes total and can answer no more than 40 problems. You want to maximize your total score. 2. **Define variables:** Let $x$ = number of geography problems. Let $y$ = number of history problems. 3. **Write constraints:** Time constraint: $1x + 5y \leq 60$ Number of problems constraint: $x + y \leq 40$ Also, $x \geq 0$, $y \geq 0$ (cannot do negative problems). 4. **Write the objective function:** Maximize total score: $S = 5x + 15y$ 5. **Solve the system:** We want to find integer values of $x$ and $y$ satisfying constraints that maximize $S$. 6. **Check corner points of the feasible region:** - When $x + y = 40$ and $x + 5y = 60$: Solve simultaneously: $$x + y = 40$$ $$x + 5y = 60$$ Subtract first from second: $$4y = 20 \Rightarrow y = 5$$ Then $x = 40 - 5 = 35$ Check if this satisfies constraints: yes. Calculate score: $$S = 5(35) + 15(5) = 175 + 75 = 250$$ - When $x + y = 40$ and $y = 0$: Then $x = 40$, time used = $40$ minutes (less than 60), score: $$S = 5(40) + 15(0) = 200$$ - When $x + 5y = 60$ and $x = 0$: Then $5y = 60 \Rightarrow y = 12$, check $x + y = 0 + 12 = 12 \leq 40$ valid. Score: $$S = 5(0) + 15(12) = 180$$ 7. **Conclusion:** The highest score is $250$ points when you do $35$ geography problems and $5$ history problems. **Final answers:** (a) $x=35$ geography problems, $y=5$ history problems. (b) Highest score = $250$ points.