A rectangular garden measures 15 meters by 20 meters. If a path of uniform width is constructed around the garden, and the total area including the path is 396 square meters, what is the width of the path? - AIKO, infinite ways to autonomy.
Discover the Hidden Math Behind Outdoor Space Expansion
Discover the Hidden Math Behind Outdoor Space Expansion
Why are homeowners and garden designers increasingly rethinking how to enhance outdoor living areas without rebuilding? A rectangular garden measuring 15 meters by 20 meters offers a compelling starting point—offering ample space that invites creative additions. Adding a uniform-width path around such a garden transforms it into a more functional and visually appealing outdoor room, especially as people seek ways to maximize small urban yards or backyard retreats. When a path of consistent width surrounds the garden, the total area grows to 396 square meters—raising a practical and intriguing question: How wide is that walkway?
This query is gaining traction across mobile devices, where users are turned on by tangible, practical problems. Combining precise geometry with real-world applications, this problem blends mathematical reasoning and everyday landscaping needs, reflecting growing interest in smart home design and outdoor productivity. The total area—396 square meters—creates a clear numerical challenge rooted in a relatable garden shape.
Understanding the Context
Understanding the expansion begins with recognizing that adding a path increases both length and width. The original garden spans 15 meters (width) by 20 meters (length). With a rectangular path uniformly placed around the perimeter, the total dimensions grow by twice the path’s width—2 w—on each side. Therefore, the new overall dimensions become (15 + 2w) meters wide and (20 + 2w) meters long.
The area formula reveals the path’s impact:
New area = (15 + 2w)(20 + 2w) = 396
Expanding this gives:
300 + 30w + 40w + 4w² = 396
4w² + 70w + 300 = 396
Subtracting 396 from both sides:
4w² + 70w – 96 = 0
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Key Insights
This quadratic equation simplifies to a manageable form for solving with the quadratic formula:
w = [–70 ± √(70² – 4×4×(–96))]/(2×4)
w = [–70 ± √(4900 + 1536)]/8
w = [–70 ± √6436]/8
Calculating the square root and simplifying yields:
w ≈ 1.1 meters (discarding the negative root, as width must be positive)
This precise calculation proves invaluable to homeowners planning renovations—offering clarity on how much space a path will consume. The 1.1-meter width balances aesthetics and functionality, inviting gentle use of walking paths, potted gardens, or outdoor seating.
This problem isn’t just a math exercise—it reflects growing trends in American home design. With urban spaces shrinking and wellness
