A rectangular prism has a surface area of 94 square units and dimensions in the ratio 1:2:3. Find the volume.
["Finding the Volume of a Rectangular Prism with Given Surface Area and Dimension Ratios", "When tackling geometry problems involving rectangular prisms, understanding the relationship between surface area, dimensions, and volume is essential. In this article, we explore a specific case: a rectangular prism with a surface area of 94 square units and dimensions in the ratio 1:2:3. We’ll determine the exact dimensions and compute the volume.", "---", "### Understanding the Problem", "A rectangular prism has six rectangular faces. Given the dimensions in the ratio 1:2:3, let the actual dimensions be:", "- Length = ( x )\n- Width = ( 2x )\n- Height = ( 3x )", "These expressions reflect the 1:2:3 proportional relationship.", "---", "### Step 1: Formula for Surface Area", "The surface area ( S ) of a rectangular prism is given by:", "[\nS = 2(lw + lh + wh)\n]", "Substituting the dimensions:", "[\nl = x,\quad w = 2x,\quad h = 3x\n]", "Compute each pairwise product:", "- ( lw = x \cdot 2x = 2x^2 )\n- ( lh = x \cdot 3x = 3x^2 )\n- ( wh = 2x \cdot 3x = 6x^2 )", "Now substitute into the surface area formula:", "[\nS = 2(2x^2 + 3x^2 + 6x^2) = 2(11x^2) = 22x^2\n]", "We’re told the surface area is 94 square units:", "[\n22x^2 = 94\n]", "---", "### Step 2: Solve for ( x^2 )", "[\nx^2 = \frac{94}{22} = \frac{47}{11}\n]", "---", "### Step 3: Find the Actual Dimensions", "Using ( x^2 = \frac{47}{11} ), we find:", "[\nx = \sqrt{\frac{47}{11}}\n]", "Now compute each dimension:", "- ( l = x = \sqrt{\frac{47}{11}} )\n- ( w = 2x = 2\sqrt{\frac{47}{11}} )\n- ( h = 3x = 3\sqrt{\frac{47}{11}} )", "---", "### Step 4: Calculate the Volume", "The volume ( V ) of a rectangular prism is:", "[\nV = l \ imes w \ imes h = x \cdot 2x \cdot 3x = 6x^3\n]", "Compute ( x^3 ):", "[\nx^3 = \left(\sqrt{\frac{47}{11}}\right)^3 = \left(\frac{47}{11}\right)^{3/2}\n]", "So,", "[\nV = 6 \left(\frac{47}{11}\right)^{3/2}\n]", "This exact expression can be simplified numerically or left in simplified radical form. Approximate:", "- ( \frac{47}{11} \approx 4.2727 )\n- ( \sqrt{4.2727} \approx 2.067 )\n- ( (4.2727)^{3/2} = 4.2727 \ imes 2.067 \approx 8.816 )", "Then:", "[\nV \approx 6 \ imes 8.816 \approx 52.9 \ ext{ cubic units}\n]", "---", "### Final Answer", "The volume of the rectangular prism is approximately 52.9 cubic units. For precision, the exact volume is:", "[\nV = 6 \left( \frac{47}{11} \right)^{3/2} \ ext{ cubic units}\n]", "---", "### Why This Problem Matters in Geometry & Applications", "Understanding how to extract real-world dimensions from ratios and surface area helps in architecture, engineering, packaging design, and manufacturing. Knowing how the surface area constrains volume allows for optimization and accurate material estimation.", "---", "---", "Keywords: rectangular prism volume, surface area formula, dimensions ratio 1:2:3, geometry problem solution, calculate prism volume, solve prism with ratios", "Meta Description:\nLearn how to find the volume of a rectangular prism with a surface area of 94 square units and dimensions in the ratio 1:2:3. Step-by-step solution with exact and approximate answers."]
