A rectangular garden is 15 meters long and 10 meters wide. A path of uniform width is to be built around the garden, increasing the total area to 286 square meters. Find the width of the path.
["Finding the Width of a Uniform Garden Path Surrounding a Rectangular Garden", "If you're planning to enhance your rectangular garden with a walking path, understanding how to calculate the path’s width based on total area can help ensure a perfectly proportioned design. This article presents a practical problem: a rectangular garden measuring 15 meters long and 10 meters wide, with a uniformly wide path added around it, increasing the total area to 286 square meters. We explore how to determine the precise width of the path.", "---", "### Garden and Path Layout", "The original garden dimensions are:\n- Length = 15 meters\n- Width = 10 meters", "A path of uniform width, say x meters, is built around the entire garden. This transforms the overall dimensions as follows:\n- Total length with path = 15 + 2x (adding x meters on each side)\n- Total width with path = 10 + 2x", "Thus, the total area including the path is:\n[\n\ ext{Total Area} = (15 + 2x)(10 + 2x)\n]\nWe are told this equals 286 square meters:", "[\n(15 + 2x)(10 + 2x) = 286\n]", "---", "### Step-by-Step Solution", "First, expand the left-hand side:", "[\n(15 + 2x)(10 + 2x) = 150 + 30x + 20x + 4x^2 = 4x^2 + 50x + 150\n]", "Set this equal to 286:", "[\n4x^2 + 50x + 150 = 286\n]", "Subtract 286 from both sides:", "[\n4x^2 + 50x - 136 = 0\n]", "Simplify the quadratic equation by dividing all terms by 2:", "[\n2x^2 + 25x - 68 = 0\n]", "---", "### Using the Quadratic Formula", "For $ax^2 + bx + c = 0$, the solutions are:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, $a = 2$, $b = 25$, $c = -68$:", "[\nx = \frac{-25 \pm \sqrt{25^2 - 4(2)(-68)}}{2 \cdot 2} = \frac{-25 \pm \sqrt{625 + 544}}{4} = \frac{-25 \pm \sqrt{1169}}{4}\n]", "Since √1169 ≈ 34.2, we get:", "[\nx = \frac{-25 + 34.2}{4} \approx \frac{9.2}{4} \approx 2.3\n]", "(The negative root is discarded since width must be positive.)", "---", "### Final Answer and Practical Insight", "The width of the path is approximately 2.3 meters. However, for precise design, using exact values:", "[\nx = \frac{-25 + \sqrt{1169}}{4}\n]", "But checking integer possibilities reveals that $x = 2$ gives total area:", "[\n(15 + 4)(10 + 4) = 19 \ imes 14 = 266 \quad (\ ext{less than 286})\n]", "$ x = 2.5 $:\n[\n(15 + 5)(10 + 5) = 20 \ imes 15 = 300 \quad (\ ext{too large})\n]", "Since $x = 2.3$ is the clean, accurate solution satisfying the 286 m² requirement, it’s ideal for landscape design.", "---", "### Why This Matters for Garden Design", "Calculating path width ensures:\n- Balanced aesthetics and usability\n- Proper distribution of materials (e.g., stone, mulch)\n- Avoiding excessive space usage", "With a rectilinear layout and uniform path, mathematical precision supports beautiful functionality.", "---", "Summary\n- Garden size: 15 × 10 m\n- Total area with path: 286 m²\n- Path width x satisfies: (15 + 2x)(10 + 2x) = 286\n- Exact solution: $ x = \frac{-25 + \sqrt{1169}}{4} \approx 2.3 $ meters", "Planning with exact square meters prevents costly mistakes and elevates garden quality.", "---", "Keywords: rectangular garden path width, uniform path around garden, garden area calculation, path math problem, garden design proportions, rectangular garden with path, how to calculate path width", "Meta Description:\nDiscover how to calculate the width of a uniform rectangular garden path that increases total area to 286 m² when the garden is 15m by 10m. Precise formula and step-by-step solution included."]
