Volume = \( l \cdot w \cdot h = 3x \cdot x \cdot 2x = 6x^3 = 108 \Rightarrow x^3 = 18 \Rightarrow x = \sqrt[3]{18} \)

["Understanding Volume Calculation: Solving for ( x ) with ( Volume = l \cdot w \cdot h = 6x^3 = 108 )", "When solving problems involving three-dimensional geometry, volume calculations are essential — especially in math education, engineering, and real-world applications. One classic example involves a rectangular prism with sides expressed in terms of a variable ( x ), resulting in a cubic equation like ( 6x^3 = 108 ). In this article, we’ll break down step-by-step how to solve for ( x ) and understand the mathematical reasoning behind finding ( x = \sqrt[3]{18} ), making volume problems clearer and more approachable.", "---", "### Volume Formula: The Foundation of the Problem", "The volume ( V ) of a rectangular prism is calculated using the formula:\n[\nV = l \cdot w \cdot h\n]", "In this particular problem, the length, width, and height are given algebraically in terms of ( x ):\n[\nl = 3x, \quad w = x, \quad h = 2x\n]", "Substituting these expressions into the volume formula gives:\n[\nV = (3x)(x)(2x)\n]", "---", "### Step 1: Simplify the Volume Expression", "Multiply the three factors:\n[\nV = 3 \cdot 1 \cdot 2 \cdot x \cdot x \cdot x = 6x^3\n]", "So,\n[\n6x^3 = 108\n]", "This equation relates the variable ( x ) directly to the given volume.", "---", "### Step 2: Solve for ( x^3 )", "To isolate ( x^3 ), divide both sides of the equation by 6:\n[\nx^3 = \frac{108}{6} = 18\n]", "Now, take the cube root of both sides to solve for ( x ):\n[\nx = \sqrt[3]{18}\n]", "---", "### Why This Matters: Real-World and Academic Applications", "Understanding how to manipulate volume equations helps students and professionals alike:", "- Engineering and Design: Calculating internal space for containers, vehicles, and machinery.\n- Architecture: Ensuring rooms fit desired volume requirements.\n- Mathematics Education: Reinforcing algebraic manipulation, roots, and cube roots in word problems.", "By breaking down complex expressions step-by-step, learners build confidence in tackling similar challenges.", "---", "### Final Result:\n[\nx = \sqrt[3]{18}\n]", "This precise solution shows how variable relationships in volume formulas enable exact answers, even when starting from simplified or scaled dimensions.", "---", "Key Takeaways:\n- Always substitute given dimensions into the volume formula.\n- Multiply factors carefully to simplify expressions.\n- Isolate the variable by dividing or applying roots.\n- Always verify your solution by plugging ( x = \sqrt[3]{18} ) back into the original volume equation.", "---", "Understanding the stepwise breakdown of volume calculations not only solves the immediate equation but strengthens mathematical reasoning for future challenges in geometry and algebra.", "---", "Keywords: ( 6x^3 = 108 ), solve for ( x ), volume formula ( V = lwh ), ( x = \sqrt[3]{18} ), cube root, three-dimensional geometry, algebra, math education."]