Let the width of the path be \( x \) meters. The dimensions of the planting area inside the path are \( (24 - 2x) \) meters by \( (15 - 2x) \) meters. The area of the planting area is given by:

["# Understanding Planting Area Dimensions: Optimizing Garden Paths", "When designing a garden with a surrounding path, careful planning of the planting zone is essential for both aesthetics and functionality. One key parameter is the width of the path, denoted by ( x ) meters. This width directly affects the available area where plants grow. Understanding how the planting area dimensions depend on ( x ) helps maximize usable garden space.", "### How the Planting Area Is Defined", "Given a path width of ( x ) meters, the total garden (including the path) forms a larger rectangle. However, the actual planting area is restricted inside this path, forming a smaller, central rectangle inside the path boundaries.", "The original outer dimensions of the garden — including the path — are:\n- Length: ( 24 ) meters\n- Width: ( 15 ) meters", "Since the path surrounds the planting zone on all sides, subtracting ( x ) meters from each side reduces the planting dimensions. Specifically:\n- The planting length is ( (24 - 2x) ) meters (losing ( x ) meters from each end)\n- The planting width is ( (15 - 2x) ) meters (losing ( x ) meters from each side)", "### The Area of the Planting Region", "The area of the planting area ( A(x) ) is the product of its length and width:", "[\nA(x) = (24 - 2x)(15 - 2x)\n]", "This expression represents a quadratic function in terms of ( x ). Expanding it provides:", "[\nA(x) = (24)(15) - 24 \cdot 2x - 15 \cdot 2x + (2x)^2 = 360 - 48x - 30x + 4x^2\n]", "[\nA(x) = 4x^2 - 78x + 360\n]", "### Maximizing the Planting Space", "Since the path width ( x ) must be positive and physically feasible (e.g., ( x < 7.5 ) to ensure positive planting dimensions), the domain is ( 0 < x < 7.5 ).", "The function ( A(x) = 4x^2 - 78x + 360 ) is a parabola opening upwards (since the coefficient of ( x^2 ) is positive), meaning it has a minimum, not a maximum. However, the goal is to maximize plantable area, so we analyze ( A(x) ) over its valid interval.", "Although the vertex lies outside the usable range, evaluating ( A(x) ) shows maximum area occurs at smaller values of ( x )—especially near ( x = 0 ) or approaching it. The largest valid planting area occurs when ( x = 0 ), giving:", "[\nA(0) = (24)(15) = 360 \ ext{ square meters}\n]", "As ( x ) increases, the product decreases due to reduced dimension loss. For example, at ( x = 3 ):", "[\nA(3) = (24 - 6)(15 - 6) = 18 \ imes 9 = 162 \ ext{ square meters}\n]", "Thus, the planting area diminishes as the path widens, emphasizing the importance of minimizing path width for larger planted spaces.", "### Practical Considerations", "- Minimizing pathway width preserves more planting area.\n- Soil depth and accessibility must balance aesthetics and usability.\n- Choosing ( x ) too large reduces planting viability.", "### Conclusion", "The planting area dimensions are defined by ( (24 - 2x) \ imes (15 - 2x) ), with area ( A(x) = 4x^2 - 78x + 360 ). While this quadratic increases initially, practical constraints limit ( x ), making minimal path width ideal. Careful selection of ( x ) ensures a functional path without sacrificing usable planting space—critical for productive and beautiful gardens.", "Optimize your garden layout today by considering both form and function: balance path width with planting area to create a harmonious, thriving landscape."]