WizardMath 7B:
Compact Mathematical AI

>>> This calculus optimization problem is confusing me: "A farmer has 200 feet of fencing to create a rectangular pen. What dimensions will maximize the area?" 🎓 *Your Calculus Tutor activates...* Excellent optimization problem! This beautifully demonstrates how calculus solves real-world problems. Let me guide you through the complete solution. 🎯 UNDERSTANDING OPTIMIZATION: We want to find the maximum area given a constraint (fixed perimeter). This is a classic "constrained optimization" problem. 📐 SETTING UP THE PROBLEM: Let's define variables: • Length = l • Width = w • Perimeter constraint: 2l + 2w = 200 • Area to maximize: A = l × w 🔍 CREATING A SINGLE-VARIABLE FUNCTION: From the constraint: 2l + 2w = 200 Simplify: l + w = 100 Solve for one variable: w = 100 - l Substitute into area formula: A(l) = l × (100 - l) = 100l - l² 🧮 FINDING THE MAXIMUM: Step 1: Take the derivative A'(l) = 100 - 2l Step 2: Set derivative equal to zero 100 - 2l = 0 2l = 100 l = 50 feet Step 3: Find the width w = 100 - l = 100 - 50 = 50 feet 🔍 VERIFICATION (Second Derivative Test): A''(l) = -2 < 0 Since the second derivative is negative, we have a maximum! ✓ 📊 FINAL VERIFICATION: • Perimeter: 2(50) + 2(50) = 200 feet ✓ • Area: 50 × 50 = 2,500 square feet ✅ ANSWER: The pen should be 50 feet × 50 feet (a square!) 🌟 DEEPER INSIGHT: For any fixed perimeter, a square always gives the maximum area. This is related to the "isoperimetric inequality" in advanced mathematics! 📈 GRAPHICAL UNDERSTANDING: The function A(l) = 100l - l² is a parabola opening downward. Its maximum occurs at the vertex, which we found at l = 50. 🔬 REAL-WORLD APPLICATIONS: • Business: Maximizing profit, minimizing cost • Engineering: Optimal design parameters • Physics: Minimizing energy, maximizing efficiency • Economics: Resource allocation Want to explore Lagrange multipliers for multi-constraint problems, or try a related rates problem? >>> _