The perimeter is \( 2(w + 3w) = 8w = 64 \) meters.
["Perimeter Calculation: How to Solve ( 2(w + 3w) = 64 ) for ( w ) – Step-by-Step Guide", "Understanding the perimeter of a shape is essential in geometry, and solving equations to find unknown dimensions makes these problems straightforward. One common scenario involves rectangles, where the perimeter formula simplifies perfectly using algebra—like in the equation:", "[\n2(w + 3w) = 64\n]", "Here’s a detailed guide to solving this perimeter problem step-by-step, along with a helpful explanation of how the perimeter formula applies.", "---", "### Understanding the Perimeter of a Rectangle", "The perimeter ( P ) of a rectangle is calculated by:", "[\nP = 2(\ ext{length} + \ ext{width})\n]", "In our case, the width is given as ( w ), and the length is ( 3w ), indicating that the length is three times the width.", "---", "### Step-by-Step Solution to ( 2(w + 3w) = 64 )", "1. Simplify the expression inside the parentheses:", "[\nw + 3w = 4w\n]", "So the equation becomes:", "[\n2(4w) = 64\n]", "2. Multiply to simplify:", "[\n8w = 64\n]", "3. Solve for ( w ):", "Divide both sides by 8:", "[\nw = \frac{64}{8} = 8 \ ext{ meters}\n]", "---", "### Finding the Full Perimeter and Dimensions", "Now that we know ( w = 8 ), we can find the full length:", "[\n\ ext{Length} = 3w = 3 \ imes 8 = 24 \ ext{ meters}\n]", "Verify the perimeter:", "[\n2(w + 3w) = 2(8 + 24) = 2 \ imes 32 = 64 \ ext{ meters}\n]", "---", "### Why This Formula Works", "This algebraic method clearly shows how perimeter formulas translate into equations. Noting that the rectangle has two widths and two lengths helps set up ( 2(w + 3w) ), while solving the simplified equation guarantees accurate results.", "---", "### Real-World Applications", "Knowing how to calculate perimeter supports many real-life tasks, from:", "- Building fencing around a garden\n- Framing a room in construction\n- Designing signs and displays", "Mastering these math fundamentals helps grasp geometry in standard measurements and practical planning.", "---", "### Summary", "- The equation ( 2(w + 3w) = 64 ) models the perimeter of a rectangle with width ( w ) and length ( 3w ).\n- Simplifying gives ( 8w = 64 ), leading to ( w = 8 \ ext{ meters} ).\n- Full dimensions: width = 8 m, length = 24 m.\n- Using algebra makes perimeter problems quick and accurate.", "---", "Key search phrases for SEO:\n- How to calculate rectangle perimeter algebraically\n- Solve ( 2(w + 3w) = 64 )\n- Find width from perimeter formula\n- Step-by-step perimeter calculation\n- Rectangle perimeter with length = 3 × width", "Optimize your geometry studies or teaching with clear, logical problem-solving—starting from perimeter equations like ( 2(w + 3w) = 64 )."]
