A rectangular box has a volume of 108 cubic meters. If the length is triple the width and the height is twice the width, what is the width of the box?
["Finding the Width of a Rectangular Box with Given Volume Dimensions", "When solving geometry problems involving the volume of a rectangular box, understanding how length, width, and height relate is key. In this article, we’ll explore a practical real-world example: determining the width of a rectangular box when its volume and dimension relationships are defined.", "### The Problem", "A rectangular box has a total volume of 108 cubic meters. The length is three times the width, and the height is twice the width. We are asked to find the width of the box.", "---", "### Step 1: Define Variables Based on Relationships", "Let the width of the box be:", "$$\n\ ext{Width} = w\n$$", "From the problem:", "- Length = $ 3w $ (triple the width)\n- Height = $ 2w $ (twice the width)", "---", "### Step 2: Use the Volume Formula", "The volume $ V $ of a rectangular box is given by:", "$$\nV = \ ext{Length} \ imes \ ext{Width} \ imes \ ext{Height}\n$$", "Substitute the expressions in terms of $ w $:", "$$\n108 = (3w) \ imes w \ imes (2w)\n$$", "Simplify the right-hand side:", "$$\n108 = 3w \cdot w \cdot 2w = 6w^3\n$$", "---", "### Step 3: Solve for $ w $", "$$\n6w^3 = 108\n$$", "Divide both sides by 6:", "$$\nw^3 = 18\n$$", "Now take the cube root of both sides:", "$$\nw = \sqrt[3]{18}\n$$", "While $ \sqrt[3]{18} $ is exact, it simplifies further as:", "$$\nw = \sqrt[3]{18} \approx 2.62~\ ext{meters}\n$$", "However, for precise academic and technical contexts, the exact form is preferred:", "$$\nw = \sqrt[3]{18}\n$$", "---", "### Final Answer", "The width of the rectangular box is:", "$$\n\boxed{\sqrt[3]{18}~\ ext{meters}} \quad (\ ext{approximately } 2.62~\ ext{m})\n$$", "---", "### Why This Matters in Real Life", "Understanding how to calculate dimensions from volume and proportional relationships is essential in fields like architecture, shipping, packaging, and construction. This method enables accurate planning and material estimation without guesswork.", "If you're tackling a physics, engineering, or design problem, mastering such volume calculations empowers you to solve complex geometry-based challenges confidently.", "---", "Keywords: rectangular box volume, finding width from volume, dimensional relationships, cube root of 18, geometry problem solution, cubic meter calculations"]
