A rectangular prism has a volume of 108 cubic meters. Its length is twice its height, and its width is 3 meters more than its height. Find the dimensions.

["Finding the Dimensions of a Rectangular Prism with Volume 108 m³: A Step-by-Step Solution", "When studying geometry, one common challenge is determining the dimensions of a three-dimensional shape when given its volume and relationships between length, width, and height. In this article, we’ll solve a realistic and insightful problem: finding the exact dimensions of a rectangular prism with a volume of 108 cubic meters, where the length is twice the height, and the width exceeds the height by 3 meters.", "---", "### The Problem: Volume and Dimensions Relationship", "Let’s define the variables based on the given relationships:", "- Volume ( V = 108 , \ ext{m}^3 )\n- Let height = ( h ) (in meters)\n- Then, length ( l = 2h )\n- And width ( w = h + 3 )", "Using the formula for the volume of a rectangular prism:", "[\nV = l \ imes w \ imes h\n]", "Substitute the known expressions:", "[\n108 = (2h) \ imes (h + 3) \ imes h\n]", "Simplify:", "[\n108 = 2h \cdot h \cdot (h + 3) = 2h^2(h + 3)\n]", "[\n108 = 2h^3 + 6h^2\n]", "Divide both sides by 2 to simplify:", "[\n54 = h^3 + 3h^2\n]", "Rewriting:", "[\nh^3 + 3h^2 - 54 = 0\n]", "We now solve this cubic equation for ( h ), which represents the physical height and must be a positive real number.", "---", "### Solving the Cubic Equation", "We look for rational roots using the Rational Root Theorem. Possible rational roots are factors of 54:\n( \pm1, \pm2, \pm3, \pm6, \pm9, \pm27, \pm54 )", "Test ( h = 3 ):", "[\n3^3 + 3(3)^2 = 27 + 27 = 54 \quad \Rightarrow \quad 54 - 54 = 0\n]", "✅ ( h = 3 ) is a root!", "So the height is ( h = 3 ) meters.", "Now compute the other dimensions:", "- Length: ( l = 2h = 2 \ imes 3 = 6 ) meters\n- Width: ( w = h + 3 = 3 + 3 = 6 ) meters", "---", "### Verifying the Solution", "Check the volume with these dimensions:", "[\nV = 6 \ imes 6 \ imes 3 = 108 , \ ext{m}^3\n]", "✔️ The volume matches the given condition.", "Also:\n- Length ( 6 = 2 \ imes 3 ) ✔️\n- Width ( 6 = 3 + 3 ) ✔️", "All constraints are satisfied.", "---", "### Conclusion: Dimensions of the Prism", "- Height = 3 meters\n- Length = 6 meters\n- Width = 6 meters", "This configuration forms a rectangular prism with a clean 6×6×3 shape, offering equal width and length — a practical and geometrically symmetric solution.", "---", "### Additional Insight for Learners", "This problem highlights how algebraic modeling can solve real-world geometry challenges. By translating word problems into equations, we transform abstract relationships into solvable math. Understanding volume relationships is essential in fields like architecture, packaging design, and engineering.", "If you enjoyed this problem, explore similar exercises with different volume targets or constraint relationships — they sharpen analytical thinking and reinforce core concepts!", "---", "Keywords: rectangular prism volume 108 m³, dimensions formula, solve volume with relations, geometry problem-solving, height width length calculation, algebraic modeling in geometry.", "---", "By clearly defining variables, setting up equations, solving step-by-step, and verifying results, you build both math skills and problem-solving confidence. Whether for homework, exams, or personal challenge, mastering this approach unlocks more complex geometric reasoning."]