A rectangular field is 50 meters longer than it is wide. If the area is 6000 m², what are the dimensions?

["Rectangular Field Dimensions: Solving the Area Puzzle", "Understanding the dimensions of a rectangular field based on its area and relationship between length and width is a common problem in applied mathematics and geometry. In this article, we’ll explore how to determine the exact size of a rectangular field when given that the field is 50 meters longer than it is wide and has an area of 6,000 square meters.", "---", "### Problem Statement\nA rectangular field has a width that is 50 meters less than its length, and its total area is 6,000 m². What are the field’s dimensions?", "---", "### Step-by-Step Solution", "Let’s define the variables:\n- Let ( w ) = width of the field in meters\n- Then, the length ( l = w + 50 ) meters", "The area ( A ) of a rectangle is given by:\n[\nA = \ ext{length} \ imes \ ext{width} = l \ imes w\n]\nSubstituting the known values:\n[\n6000 = (w + 50) \ imes w\n]", "---", "### Expand and Rearrange the Equation", "[\nw(w + 50) = 6000\n]\n[\nw^2 + 50w - 6000 = 0\n]", "This is a quadratic equation in standard form:\n[\nw^2 + 50w - 6000 = 0\n]", "---", "### Solving the Quadratic Equation", "Use the quadratic formula:\n[\nw = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nFor our equation ( w^2 + 50w - 6000 = 0 ), coefficients are:\n- ( a = 1 )\n- ( b = 50 )\n- ( c = -6000 )", "Calculate the discriminant:\n[\n\Delta = b^2 - 4ac = 50^2 - 4(1)(-6000) = 2500 + 24000 = 26500\n]", "Now compute the square root:\n[\n\sqrt{26500} \approx 162.63\n]", "Apply the quadratic formula:\n[\nw = \frac{-50 \pm 162.63}{2}\n]", "We discard the negative solution since width cannot be negative:\n[\nw = \frac{-50 + 162.63}{2} = \frac{112.63}{2} \approx 56.31 , \ ext{meters}\n]", "Now find the length:\n[\nl = w + 50 = 56.31 + 50 = 106.31 , \ ext{meters}\n]", "---", "### Final Dimensions", "- Width: approximately 56.31 meters\n- Length: approximately 106.31 meters", "Together, these satisfy:\n- Length is 50 meters longer than width: ( 106.31 - 56.31 = 50 ) m ✅\n- Area: ( 56.31 \ imes 106.31 \approx 6000 ) m² ✅", "---", "### Why This Matters", "Accurately determining the dimensions of land based on area and side relationships is crucial for farmers, construction planners, and land surveyors. This problem reinforces key algebraic skills—translating word problems into equations, solving quadratics, and verifying results—essential for real-world applications.", "---", "### Summary", "Given a rectangular field where the length exceeds the width by 50 meters and the area is 6,000 m², solving algebraically reveals the dimensions are approximately:", "Width ≈ 56.31 meters,\nLength ≈ 106.31 meters", "Use these values to plan fencing, planting areas, or layout design confidently.", "---", "Keywords: rectangular field dimensions, solve quadratic area problem, width and length rectangle, area of rectangle solved algebraically, real-world geometry application", "Meta Description: Learn how to calculate the dimensions of a rectangular field that is 50 meters longer than it is wide, given the area is 6,000 m² — step-by-step solution with practical application."]