Let \(x\) be the width of the path. The new dimensions are \((15 + 2x)\) and \((10 + 2x)\).

["Optimizing Path Design: Maximizing Area with Variable Width (x)", "When designing pathways in landscaping, architecture, or urban planning, the dimensions of the path directly influence its functionality, usability, and integration into the surrounding environment. Let (x) represent the width of the path, a critical variable that shapes both the footprint and the accessibility of the space. By expressing the new dimensions as ((15 + 2x)) and ((10 + 2x)), one unlocks a dynamic approach to path design that adapts precisely to site constraints and design goals.", "### Understanding the Dimensions", "The original path dimensions (before adding width (x)) are typically taken to reflect the base area or existing layout. When expanding each dimension by (2x)—one unit in each direction—we create a uniform strip expansion symmetrically around the original perimeter. This model effectively models width-increasing path construction, ensuring smooth expansion with balanced growth on all sides.", "- New Length: (15 + 2x)\n- New Width: (10 + 2x)", "This notation reflects that the path extends (x) units outward from its original boundary, uniformly along both length and width.", "### Calculating Area and Functional Impact", "The total area (A(x)) of the expanded path is a key metric:", "[\nA(x) = (\ ext{Length}) \ imes (\ ext{Width}) = (15 + 2x)(10 + 2x)\n]", "Expanding this expression:", "[\nA(x) = 150 + 30x + 20x + 4x^2 = 4x^2 + 50x + 150\n]", "This quadratic function reveals how the path area increases quadratically with (x), emphasizing that even modest increases in width significantly expand the usable or constructed surface. For example:", "- At (x = 2): (A(2) = 4(4) + 50(2) + 150 = 166) square units\n- At (x = 5): (A(5) = 4(25) + 250 + 150 = 425) square units", "This growth pattern supports the design principle of scalable space—larger paths offer more flexibility in materials, drainage, and user comfort.", "### Optimizing Path Width for Practical Needs", "Choosing the right (x) involves balancing aesthetics, cost, drainage, and access:", "- Drainage: Wider paths accommodate better water runoff and reduce puddling, improving longevity.\n- Accessibility: Wider, uniform paths suit wheelchairs, strollers, and heavy wheelchair turnover.\n- Material and Cost: Increasing width means more material, so align (x) with budget constraints.\n- Functionality: Footpaths may need more space than narrow trails; integrate lighting, seating, or landscaping during expansion.", "### Applications in Real-World Settings", "- Landscaping: A path doubling garden edges encourages easier maintenance and connects spaces seamlessly.\n- Architecture: Modern buildings often use uniform pathway widths to maintain symmetry and accessibility.\n- Urban Planning: Expanding sidewalks by (x) accommodates increased pedestrian flow without reconfiguring underlying infrastructure.", "### Conclusion", "Letting (x) be the path width transforms static measurements into a strategic tool for responsive design. By modeling path expansion with ((15 + 2x)) and ((10 + 2x)), designers and planners gain granular control over spatial parameters—maximizing utility, enhancing aesthetics, and adapting to evolving project needs. Embrace variable dimensions to build paths that are not just functional, but thoughtfully integrated into their environment.", "---", "Keywords: path width optimization, expandable path dimensions, landscape design area calculation, quadratic path area formula, universal accessibility design, sustainable pathway planning"]