A rectangular garden measures 30 meters in length and 20 meters in width. A path of uniform width is to be built around the garden. If the area of the path is 200 square meters, what is the width of the path?

["Designing Serene Outdoor Spaces: The Hidden Math Behind Garden Pathways", "Wondering how to balance beauty and function in a home garden? A common challenge is integrating a circular, rectangular, or structured garden with a surrounding walkway. One frequently discussed design question involves a rectangular garden measuring 30 meters by 20 meters, with a uniform-width path surrounding it, forming a larger rectangular border. For a path area of 200 square meters, what dimensions and width create both visual harmony and practical layout? This query isn’t just about math—it reflects a growing interest in smart, intentional outdoor living spaces across the U.S. More homeowners are seeking efficient, aesthetically balanced designs that invite relaxation and connectivity with nature.", "---", "### Why searchers are asking: The Trend Behind Pathway Calculations", "Garden layout inspires active engagement on mobile devices, and paths are central to both form and function. With rising interest in outdoor living, sustainable planning, and smart home integration, many users now seek precise measurements to optimize garden footprints. A rectangular garden design with uniform pathways creates predictable space usage—expanding useful areas while enhancing curb appeal. The growing popularity of platforms like Pinterest, Instagram, and lifestyle blogs has amplified demand for relatable, accurate design guidance, making questions about precise path width and garden footprint especially relevant in current digital trends.", "---", "### The Math Behind the Pathway: A Clear Explanation", "The garden measures 30 meters by 20 meters. Adding a uniform path of width \( x \) meters around it increases both length and width by \( 2x \), forming a larger rectangle of dimensions (30 + 2x) by (20 + 2x). The area of just the path is 200 square meters, which equals:", "\[\n\ ext{Path Area} = (\ ext{Larger Area}) - (\ ext{Garden Area}) = (30 + 2x)(20 + 2x) - (30 \ imes 20)\n\]", "Expanding: \n\[\n(30 + 2x)(20 + 2x) = 600 + 60x + 40x + 4x^2 = 600 + 100x + 4x^2\n\] \nSubtracting base garden area: \n\[\n600 + 100x + 4x^2 - 600 = 100x + 4x^2\n\] \nSet equal to 200: \n\[\n4x^2 + 100x = 200\n\] \nSimplify: \n\[\n4x^2 + 100x - 200 = 0\n\]", "Dividing the equation by 4: \n\[\nx^2 + 25x - 50 = 0\n\]", "---", "### Solving for the Path Width", "Using the quadratic formula: \n\[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\quad a=1, b=25, c=-50\n\] \n\[\nx = \frac{-25 \pm \sqrt{625 + 200}}{2} = \frac{-25 \pm \sqrt{825}}{2}\n\] \n\(\sqrt{825} \approx 28.7\) \n\[\nx = \frac{-25 + 28.7}{2} \approx 1.85,\quad \ ext{discard negative root}\n\]", "Thus, the path width is approximately 1.85 meters—a realistic, proportional estimate that balances usability and space efficiency.", "---", "### Common Questions About Path Dimensions", "Q: How is the path width calculated around a rectangular garden? \nA: Start with original garden area (30 m × 20 m = 600 m²), add the path area (200 m²), forming a larger rectangle. Use the difference in dimensions (increased length and width) to solve a quadratic equation based on total area.", "Q: Can the path width vary around corners in a rectangular garden? \nA: The scenario assumes uniform width, simplifying math and ensuring symmetry—common in standard landscaping practice.", "Q: How does pathway width affect garden usability? \nA: A 1.85-meter path allows comfortable walking while leaving room for garden beds, seating, and greenery, offering both form and function.", "---", "### Opportunities and Design Considerations", "Benefits \n- Enhanced mobility through outdoor living areas \n- Improved definition of garden boundaries \n- Better integration with modern, low-maintenance landscaping", "Challenges \n- Balancing expansion with property lines \n- Competing priorities like plant placement and sunlight access \n- Long-term maintenance considerations related to drainage and soil coverage", "Realistic Expectations \nA 1.8-meter path offers proportional space without overwhelming the garden footprint. It’s ideal for enjoying nature consecutively while preserving imagination for future garden evolution.", "---", "### Common Misconceptions About Garden Paths", "Many believe larger paths improve beauty, but excessive expansion reduces planting space and increases maintenance. Others assume uniform width is impractical—yet precise calculations enable harmonious, usable layouts. Accurate measurements preserve both functionality and aesthetic intent.", "---", "### Who Benefits from This Knowledge?", "This guide supports homeowners planning expansions, urban gardeners reimagining balconies or backyards, and sustainability-focused designers aiming for efficient land use. Whether building in a suburban lot or a city balcony garden, understanding path math ensures thoughtful, lasting outdoor spaces.", "---", "### A Gentle Call to Curiosity", "While precise math brings clarity, good design begins with curiosity—about how space shapes experience, how small choices amplify comfort, and how outdoor environments quietly nurture well-being. Exploring measurements like path width is not about perfection, but about crafting spaces where life feels intentional, calm, and deeply connected to nature. Let this guide spark thoughtful choices that last."]