A rectangle's length is 3 times its width. If the perimeter is 64 cm, find the dimensions of the rectangle.
["Finding the Dimensions of a Rectangle: Length Three Times the Width with a 64 cm Perimeter", "When solving geometry problems, one of the most practical applications involves rectangles—shapes you encounter frequently in everyday life, from frames to rooms. Today, we’ll explore a specific rectangle problem: when the length is three times the width, and the perimeter is 64 cm. Understanding how to derive the dimensions step by step not only sharpens algebraic thinking but also helps in real-world measurements and design.", "### Understanding the Problem", "We’re told two key facts about the rectangle:", "- The length (L) is 3 times the width (W):\n [\n L = 3W\n ]", "- The perimeter of the rectangle is 64 cm:\n [\n \ ext{Perimeter} = 64,\ ext{cm}\n ]", "The perimeter ( P ) of a rectangle is calculated using the formula:\n[\nP = 2L + 2W\n]", "### Substitute and Solve Using Algebra", "We substitute ( L = 3W ) into the perimeter formula:", "[\n64 = 2(3W) + 2W\n]", "Now, simplify:", "[\n64 = 6W + 2W\n]\n[\n64 = 8W\n]", "Now, solve for ( W ):", "[\nW = \frac{64}{8} = 8,\ ext{cm}\n]", "Now find the length ( L ):", "[\nL = 3W = 3 \ imes 8 = 24,\ ext{cm}\n]", "### Final Answer and Dimensions", "- Width (W): 8 cm\n- Length (L): 24 cm", "To verify, compute the perimeter with these dimensions:\n[\n2L + 2W = 2(24) + 2(8) = 48 + 16 = 64,\ ext{cm} \quad \ ext{✓}\n]", "### Why This Matters", "Knowing how to calculate rectangle dimensions when one side relates directly to the other is essential in architecture, interior design, engineering, and even wireframe modeling. This particular ratio (3:1) appears often in design elements requiring proportional scaling.", "### Summary", "Given:\n- ( L = 3W )\n- ( P = 64,\ ext{cm} )", "Dimensions:\n- Width = 8 cm\n- Length = 24 cm", "[\n\boxed{\ ext{Width:} ~8,\ ext{cm},\ \ ext{Length:} ~24,\ ext{cm}}\n]", "If you're tasked with solving similar problems—whether estimating material needs for a project or drafting design plans—this foundational approach using algebra and the perimeter formula will serve you well."]
