A rectangular garden has a length 5 meters more than twice its width. If the area is 150 square meters, find the dimensions.
["Rectangular Garden Dimensions: Solving for Length and Width Given Area and Relationship", "Creating a well-proportioned rectangular garden begins with understanding the mathematical relationships between its length, width, and area. In this article, we’ll solve a common problem involving a rectangular garden where the length is defined in relation to the width, and the total area is given as 150 square meters. If the garden’s length is 5 meters more than twice its width, how do we find the actual dimensions?", "---", "### Understanding the Problem", "We’re told:", "- The garden is rectangular.\n- The length $ L $ is 5 meters more than twice the width $ W $:\n $ L = 2W + 5 $\n- The area of the garden is 150 square meters:\n $ \ ext{Area} = L \ imes W = 150 $", "This gives us a straightforward equation to solve:\nSubstitute $ L = 2W + 5 $ into the area formula.", "---", "### Step-by-Step Solution", "1. Substitute the expression for $ L $ into the area formula:\n $$\n (2W + 5) \ imes W = 150\n $$", "2. Expand the equation:\n $$\n 2W^2 + 5W = 150\n $$", "3. Rearrange into standard quadratic form:\n $$\n 2W^2 + 5W - 150 = 0\n $$", "4. Solve the quadratic equation\n Use the quadratic formula:\n $$\n W = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n $$\n Here, $ a = 2 $, $ b = 5 $, $ c = -150 $. Plug in the values:\n $$\n W = \frac{-5 \pm \sqrt{5^2 - 4(2)(-150)}}{2(2)} = \frac{-5 \pm \sqrt{25 + 1200}}{4} = \frac{-5 \pm \sqrt{1225}}{4}\n $$", "5. Calculate the square root:\n $$\n \sqrt{1225} = 35\n $$", "6. Find the two possible solutions:\n $$\n W = \frac{-5 + 35}{4} = \frac{30}{4} = 7.5 \quad \ ext{(valid, width must be positive)}\n $$\n $$\n W = \frac{-5 - 35}{4} = \frac{-40}{4} = -10 \quad \ ext{(invalid, width cannot be negative)}\n $$", "7. Determine the length using the valid width:\n $$\n L = 2W + 5 = 2(7.5) + 5 = 15 + 5 = 20 \ ext{ meters}\n $$", "---", "### Final Dimensions", "- Width: 7.5 meters\n- Length: 20 meters", "This rectangular garden has a perfect balance where the length exceeds the width by 12.5 meters — consistent with the statement that it’s 5 meters more than twice the width ($ 2 \ imes 7.5 + 5 = 20 $).", "---", "### Practical Applications", "Understanding how to model real-world problems with algebra and geometry helps in gardening design, landscaping, and even spatial planning. By clearly defining relationships and applying quadratic equations, homeowners and professionals can efficiently calculate optimal growing spaces.", "---", "### Conclusion", "Solving for the dimensions of a rectangular garden using the given length-width relationship and known area is a practical exercise in algebra. With a length 5 meters more than twice the width and an area of 150 square meters, the correct dimensions are width = 7.5 meters and length = 20 meters. Use this method anytime you need to combine geometric formulas and linear relationships in garden planning or related design work.", "---", "Keywords: rectangular garden dimensions, solve rectangle area problem, find garden width and length, quadratic equation garden problem, garden planning equations, 5 meters more than twice width\nMeta Description: Learn how to find the dimensions of a rectangular garden where the length is 5 meters more than twice the width, using algebra and area formulas. Solve step-by-step with a real-life gardening example."]
