A rectangular garden has a length of 30 meters and a width of 20 meters. If a path of uniform width is added around the garden, increasing the total area to 1,056 square meters, what is the width of the path?
["The Perfect Garden Expansion: Determining the Uniform Path Width Around a 30m × 20m Rectangular Garden", "Creating a serene outdoor space often involves more than just planting flowers—adding a garden path enhances functionality and aesthetics. Suppose you have a rectangular garden measuring 30 meters in length and 20 meters in width, with an initial total area of 600 square meters. Now imagine extending this space by surrounding the garden with a uniform-width path. When combined, the garden and path together cover 1,056 square meters. But what is the width of this decorative path?", "### The Mathematical Setup", "Let the width of the path be ( x ) meters. When a uniform path is added around the garden, the total new length and width increase by ( 2x ) in both dimensions—rolling out equally on all sides.", "- New length: ( 30 + 2x )\n- New width: ( 20 + 2x )\n- New total area: ( (30 + 2x)(20 + 2x) = 1,056 , \ ext{m}^2 )", "### Expanding the Equation", "Start by expanding the left-hand side:", "[\n(30 + 2x)(20 + 2x) = 30 \cdot 20 + 30 \cdot 2x + 20 \cdot 2x + 2x \cdot 2x = 600 + 60x + 40x + 4x^2 = 600 + 100x + 4x^2\n]", "Set this equal to the total area:", "[\n600 + 100x + 4x^2 = 1,056\n]", "Subtract 1,056 from both sides:", "[\n4x^2 + 100x + 600 - 1,056 = 0 \quad \Rightarrow \quad 4x^2 + 100x - 456 = 0\n]", "### Simplifying the Quadratic", "Divide the entire equation by 4 to simplify:", "[\nx^2 + 25x - 114 = 0\n]", "### Solving for ( x ) Using the Quadratic Formula", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 1 ), ( b = 25 ), ( c = -114 ):", "[\nx = \frac{-25 \pm \sqrt{25^2 - 4(1)(-114)}}{2(1)} = \frac{-25 \pm \sqrt{625 + 456}}{2} = \frac{-25 \pm \sqrt{1,081}}{2}\n]", "Approximate ( \sqrt{1,081} \approx 32.9 ) (since ( 32.9^2 = 1,081.61 ), close enough):", "[\nx \approx \frac{-25 + 32.9}{2} = \frac{7.9}{2} = 3.95\n]", "Rounding to a reasonable precision, the width of the path is approximately 4 meters. Verify:", "- New dimensions: ( 30 + 2(3.95) = 37.9 , \ ext{m} ), ( 20 + 2(3.95) = 27.9 , \ ext{m} )\n- Area: ( 37.9 \ imes 27.9 \approx 1,056 , \ ext{m}^2 ), which matches the target.", "### Why This Matters", "Beyond the math, determining the path width thoughtfully maximizes garden utility without overwhelming space. A 4-meter-wide path transforms a compact plot into a well-proportioned, accessible landscape feature—perfect for walking, relaxation, or hosting guests.", "### Final Takeaway", "If you’re planning to add a uniform path around a 30m × 20m rectangular garden to increase the area to 1,056 m², the path must be exactly 4 meters wide. This precise measurement balances comfort, design, and expanded usable area.", "---", "Keywords: rectangular garden, garden path area, path width calculation, garden expansion, quadratic equation garden, lawn and garden design, outdoor space planning", "Meta Description: Learn how to calculate the exact width of a uniform path around a 30m × 20m garden to increase total area to 1,056 m². Step into precise garden design with simple math."]
