1. **Problem Statement:** Nancy Lerner has 1,200 minutes to study for two exams. Her overall course score is the minimum of the two exam scores. She wants to maximize this minimum score. 2. **Score Formulas:** - Score on Exam 1: $x = \frac{\text{minutes on Exam 1}}{20}$ - Score on Exam 2: $y = \frac{\text{minutes on Exam 2}}{30}$ 3. **Budget Constraint:** Total study time is 1,200 minutes: $$20x + 30y = 1200$$ This is the "budget line" showing all combinations of scores $(x,y)$ achievable. 4. **Rewrite Budget Line:** $$y = \frac{1200 - 20x}{30} = 40 - \frac{2}{3}x$$ This line has intercepts at $x=60$ (if all time on Exam 1) and $y=40$ (if all time on Exam 2). 5. **Indifference Curves:** Nancy's overall score is $\min(x,y)$, so her indifference curves are L-shaped with kinks along the line $x=y$. - For a given overall score $k$, the indifference curve is: $$\min(x,y) = k \implies x \ge k, y \ge k$$ This forms an L-shape with a corner at $(k,k)$. 6. **Kink Line:** The kinks of the indifference curves lie on the line: $$x = y$$ 7. **Find Point A:** Point A is where the kink line $x=y$ intersects the budget line: Set $y = x$ in budget line: $$x = 40 - \frac{2}{3}x$$ $$x + \frac{2}{3}x = 40$$ $$\frac{5}{3}x = 40$$ $$x = \frac{40 \times 3}{5} = 24$$ So, $x = y = 24$. 8. **Interpretation:** At point A, Nancy allocates study time so that both exam scores are equal at 24 points, maximizing her minimum score. 9. **Draw Nancy's Indifference Curve through A:** This curve corresponds to $\min(x,y) = 24$ with a kink at $(24,24)$. **Summary:** - Budget line: $$20x + 30y = 1200$$ - Kink line: $$x = y$$ - Point A: $$(24,24)$$ - Indifference curve through A: L-shaped with corner at $(24,24)$ This setup helps Nancy maximize her minimum exam score by balancing study time.