#### Width: \( \frac{25}{3} \) m, Length: \( \frac{65}{3} \) m1. A rectangular garden is 15 meters long and 10 meters wide. If a path of uniform width is built inside the garden along the perimeter, reducing the area available for planting by 40 square meters, what is the width of the path?

["### How Wide Is the Path Inside the Garden Reducing Planting Area by 40 m²?\nA rectangular garden is 15 meters long and 10 meters wide. A uniform-width path runs along the inside perimeter, reducing the planting area by 40 square meters. What is the width of the path?", "---", "#### Understanding the Problem", "We start with a rectangular garden measuring:\n- Length (L) = 15 meters\n- Width (W) = 10 meters", "The original planting area is:\n[\n\ ext{Area}{\ ext{original}} = 15 \ imes 10 = 150 \ ext{ m}^2\n]", "A path of uniform width ( x ) meters is built inside the garden along all four sides. This path reduces the available planting area by 40 m², so the new planting area becomes:\n[\n\ ext{Area}^2}} = 150 - 40 = 110 \ ext{ m\n]", "We need to find the value of ( x ), the width of the path.", "---", "#### Dimensions of the Planting Area After the Path", "Because the path runs uniformly along the inside edges, it reduces both the length and width of the planting area:", "- Reduced length = ( 15 - 2x ) (path cuts 2×( x ) from both ends)\n- Reduced width = ( 10 - 2x )", "Therefore, the planting area is:\n[\n(15 - 2x)(10 - 2x)\n]", "According to the problem, this area equals 110 m²:\n[\n(15 - 2x)(10 - 2x) = 110\n]", "---", "#### Expanding the Equation", "Multiply the left-hand side:\n[\n(15 - 2x)(10 - 2x) = 150 - 30x - 20x + 4x^2 = 150 - 50x + 4x^2\n]", "So the equation becomes:\n[\n4x^2 - 50x + 150 = 110\n]", "Subtract 110 from both sides:\n[\n4x^2 - 50x + 40 = 0\n]", "---", "#### Simplifying the Quadratic Equation", "Divide all terms by 2 to simplify:\n[\n2x^2 - 25x + 20 = 0\n]", "Use the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nwhere ( a = 2 ), ( b = -25 ), ( c = 20 )", "Calculate discriminant:\n[\n\Delta = (-25)^2 - 4(2)(20) = 625 - 160 = 465\n]", "So,\n[\nx = \frac{25 \pm \sqrt{465}}{4}\n]", "Approximate ( \sqrt{465} ):\n( \sqrt{465} \approx 21.56 )", "Then,\n[\nx = \frac{25 - 21.56}{4} = \frac{3.44}{4} = 0.86 \quad \ ext{(acceptable solution, positive and smaller than half the smaller dimension)}\n]\nor\n[\nx = \frac{25 + 21.56}{4} = \frac{46.56}{4} = 11.64 \quad \ ext{(too large — exceeds half of either dimension)}\n]", "Only ( x \approx 0.86 ) meters is physically meaningful.", "---", "#### Exact Answer — Simplifying ( \frac{25 - \sqrt{465}}{4} )", "However, toward a clean final answer, let’s verify if a simplified radical form is preferred. Since 465 factors as ( 465 = 3 \ imes 5 \ imes 31 ), it has no perfect square factor — so ( \sqrt{465} ) is simplified.", "But we can express the exact width of the path as:\n[\nx = \frac{25 - \sqrt{465}}{4} \ ext{ meters}\n]", "Numerically, this is approximately ( 0.86 ) m, but since the problem involves geometric precision and avoids rounding, the precise width is:", "[\n\boxed{ \frac{25 - \sqrt{465}}{4} } \ ext{ meters}\n]", "---", "#### Why This Width Works", "- Path width ≤ 5 m (since original width is 10 m, half is 5)\n- Our solution ( x \approx 0.86 ) m satisfies this\n- The planted area:\n [\n (15 - 2 \ imes 0.86)(10 - 2 \ imes 0.86) = (13.28)(8.28) \approx 110 \ ext{ m}^2\n ]\n confirms validity.", "---", "#### Final Notes", "- Always check if reduced dimensions remain positive:\n ( 15 - 2x > 0 \Rightarrow x < 7.5 ) — valid\n ( 10 - 2x > 0 \Rightarrow x < 5 ) — most valid solutions satisfy this\n- The uniform path width is well-defined by this constraint and area reduction.", "In summary: a path of approximately 0.86 meters (exactly ( \frac{25 - \sqrt{465}}{4} ) m) surrounds the garden, reducing planting area by 40 m².", "---", "Keywords: rectangular garden path width, path inside rectangle, area reduction by uniform border, solve for path width, garden dimensions calculation, quadratic path equation\nSEO meta description:\nDiscover the exact width of a uniform path inside a 15×10 m rectangular garden that reduces planting area by 40 m² — solved via algebra and verified with quadratic methods. Full step-by-step."]