Solution: Let the length and width of the rectangle be $ l $ and $ w $, respectively. The area is $ lw = 100 $, and the perimeter is $ P = 2(l + w) $. To minimize $ P $, the rectangle should be a square (by the AM-GM inequality or calculus). So, set $ l = w $:

["Why Optimizing for Minimal Perimeter Matters in Modern Design and Efficiency", "In a world where space, cost, and efficiency drive decision-making, the relationship between area and perimeter reveals deeper insights into design logic—particularly in rectangles, a shape familiar to everyone from architects to DIY hobbyists. When faced with a fixed area—say, 100 square units—what’s the most efficient shape to minimize fencing, material, or construction effort? The answer emerges clearly: a square. When length equals width, minimum perimeter is achieved, rooted in mathematical principles like the AM-GM inequality and confirmed by calculus. This concept transcends abstract geometry—it influences how patterns, spaces, and systems are optimized across industries.", "The connection isn’t just mathematical; it reflects a broader trend toward efficiency and resource optimization, especially in modern infrastructure and digital design. Whether arranging furniture, planning room layouts, or designing functional layouts, minimizing perimeter while maximizing area offers tangible benefits. This idea resonates with US audiences navigating tight budgets, space constraints, and a cultural emphasis on smart choices.", "How Does This Apply to Real-World Decisions?", "When $ lw = 100 $, minimizing $ P = 2(l + w) $ demands $ l = w $, making the rectangle a square. Beyond theory, this principle shapes practical choices. For example, maximizing usable space within fixed building footprints reduces material use and long-term costs. Similarly, optimizing layout efficiency in office spaces, homes, or retail environments means more effective use of floor area without expansion. Even in digital interfaces, optimizing layout ratios can enhance functionality and user experience—a quiet echo of geometric principles.", "Common Questions About Minimal Perimeter Optimization", "Q: Why is a square the most efficient rectangle for a given area? \nA: Because, mathematically, arranging equal length and width minimizes total boundary length for that fixed area. This balance avoids unnecessary space waste, reducing costs and material use.", "Q: Does this apply beyond physical spaces? \nA: Yes. In data organization, network design, and even time management, optimizing for minimal "perimeter"—the outer cost of constraint—creates more efficient systems and clearer workflows.", "Opportunities and Considerations", "Embracing the square as optimal offers clear advantages: lower material and shipping costs, easier coordination of resources, and reduced spatial waste. However, in real-world design, trade-offs emerge—structural flexibility, aesthetic preferences, or contextual building regulations may occasionally favor non-square shapes. Understanding the mathematical ideal helps balance innovation with practicality.", "Common misunderstandings abound. One myth is that all rectangles behave the same under fixed area constraints. In reality, the square offers a unique advantage. Another is overgeneralizing the principle—to a shape isn’t always better, but when efficiency and equality of sides matter, the square shines. Recognizing this nu"]