A rectangular garden measuring 24 feet by 32 feet is surrounded by a uniform path of width x feet. If the total area including the path is 1400 square feet, what is the width x of the path?

Tue, 05 May 2026 17:21:52 +0000

**Curious gardeners and homeowners across the U.S. are increasingly exploring creative ways to enhance outdoor spaces—especially when blending functionality with aesthetic appeal. One common challenge involves expanding a garden’s hit-or-miss boundaries with a thoughtfully designed path. A rectangular garden measuring 24 feet by 32 feet, enhanced by a uniform gravel or paved path of width *x* feet, offers both visual balance and practical value. With the total area—including the surrounding path—reaching 1400 square feet, many seek the precise width that transforms this layout effectively.** A rectangular garden measuring 24 feet by 32 feet, surrounded by a uniform path of width *x* feet, if the total area including the path is 1400 square feet, what is the width *x*? Designers and homeowners alike are drawn to geometric consistency—savage square or rectangular layouts framed by a steady border of material. This path not only frames the space cleanly but also elevates curb appeal and guides movement through garden zones. With the garden itself occupying 24 × 32 = 768 square feet, the path area becomes 1400 – 768 = 632 square feet—information critical for planning material needs and spatial harmony. Factoring in that the path surrounds the garden uniformly (adding *x* feet on all sides), the outer dimensions become (24 + 2x) by (32 + 2x), making the total area (24 + 2x)(32 + 2x) = 1400 square feet. This equation reveals the core challenge: solving for *x* in a real-world scenario grounded in everyday landscaping plans. How A rectangular garden measuring 24 feet by 32 feet is surrounded by a uniform path of width x feet. If the total area including the path is 1400 square feet, what is the width x of the path?
This setup forms a quadratic equation: (24 + 2x)(32 + 2x) = 1400. Expanding gives 2x(56 + 4x) + 768 = 1400 → 8x² + 224x + 768 = 1400 → 8x² + 224x – 632 = 0. Dividing through by 8 simplifies it to x² + 28x – 79 = 0. Using the quadratic formula: x = [–28 ± √(28² + 4×79)] / 2 = [–28 ± √(784 + 316)] / 2 = [–28 ± √1100] / 2. √1100 ≈ 33.17 → x = (–28 + 33.17)/2 ≈ 2.585 feet. This moderate width balances space and cost, fitting typical residential garden goals.