An industrial designer develops a biodegradable packaging box in the shape of a rectangular prism with dimensions in the ratio 2:3:5. If the volume is 1080 cm³, what is the surface area in cm²?

["Developing Sustainable Packaging: The Science Behind a Biodegradable Rectangular Prism Box", "In today’s environmentally conscious market, industrial designers are pioneering innovative solutions that balance functionality with sustainability. One inspiring example is the development of a biodegradable packaging box shaped as a rectangular prism—designed not only for protective efficiency but also for ecological responsibility.", "A recent breakthrough in this field involves an industrial designer who crafted a packaging box with dimensions in the ratio 2:3:5, forming a precise rectangular prism. This thoughtful geometric choice ensures optimal use of material and space, while maintaining structural integrity and scalability. The designer’s goal was clear: create a lightweight, compostable packaging solution that reduces plastic waste without compromising durability.", "---", "### Dimension Calculation Based on Ratio", "Given the volume of the box is 1080 cm³ and the side lengths follow the ratio 2 : 3 : 5, let’s define the dimensions as:", "- $ 2x $\n- $ 3x $\n- $ 5x $", "Step 1: Set up the volume equation", "$$\n\ ext{Volume} = (2x)(3x)(5x) = 30x^3\n$$", "$$\n30x^3 = 1080\n$$", "$$\nx^3 = \frac{1080}{30} = 36\n\quad \Rightarrow \quad\nx = \sqrt[3]{36} \approx 3.30\n$$", "But to keep the design precise and maintain exact ratios, we solve:", "$$\nx^3 = 36 \Rightarrow x = \frac{\sqrt[3]{36}}{1}\n$$", "However, since dimensions should be clean and likely rational, let’s test integer approximations or find exact forms. But instead, we proceed with exact symbolic computation:", "$$\nx = \sqrt[3]{36},\quad \ ext{so:} \quad 2x = 2\sqrt[3]{36},\quad 3x = 3\sqrt[3]{36},\quad 5x = 5\sqrt[3]{36}\n$$", "But for practical building and surface area calculation, we keep $ x $ as a variable and compute symbolically.", "---", "### Step 2: Express surface area formula", "The surface area $ S $ of a rectangular prism is:", "$$\nS = 2(lw + lh + wh)\n$$", "Substitute $ l = 2x $, $ w = 3x $, $ h = 5x $:", "$$\nS = 2\left( (2x)(3x) + (2x)(5x) + (3x)(5x) \right)\n= 2\left( 6x^2 + 10x^2 + 15x^2 \right)\n= 2(31x^2) = 62x^2\n$$", "Now solve for $ x^2 $ from $ 30x^3 = 1080 $:", "$$\nx^3 = 36 \quad \Rightarrow \quad x^2 = \frac{36}{x}\n$$", "But better: from $ x^3 = 36 \Rightarrow x = 36^{1/3} $, so:", "$$\nx^2 = (36^{1/3})^2 = 36^{2/3}\n$$", "Now compute:", "$$\nS = 62 \cdot 36^{2/3}\n$$", "We now compute $ 36^{2/3} $:", "$$\n36 = 6^2 = (2 \cdot 3)^2 = 2^2 \cdot 3^2\n\Rightarrow 36^{2/3} = (2^2 \cdot 3^2)^{2/3} = 2^{4/3} \cdot 3^{4/3}\n= \left(2^4 \cdot 3^4\right)^{1/3} = \sqrt[3]{16 \cdot 81} = \sqrt[3]{1296}\n$$", "But numerically:", "$$\n36^{1/3} \approx 3.3019 \Rightarrow (36^{1/3})^2 \approx 10.897\n$$", "$$\nS = 62 \ imes 10.897 \approx 674.6 , \ ext{cm}^2\n$$", "But let’s find an exact expression.", "Note: $ x^3 = 36 \Rightarrow x = 36^{1/3} $, so $ x^2 = 36^{2/3} $", "But from $ 30x^3 = 1080 $, $ x^3 = 36 $, so:", "Let’s express surface area exactly:", "$$\nS = 62x^2 = 62 \cdot \frac{36}{x} = 62 \cdot \frac{36}{36^{1/3}} = 62 \cdot 36^{2/3}\n$$", "Now, $ 36 = 6^2 $, so:", "$$\n36^{2/3} = (6^2)^{2/3} = 6^{4/3} = 6 \cdot 6^{1/3}\n\Rightarrow S = 62 \cdot 6 \cdot 6^{1/3} = 372 \cdot \sqrt[3]{6}\n$$", "But this is not a clean number. Let's recheck: is there an integer solution?", "Try solving $ 30x^3 = 1080 \Rightarrow x^3 = 36 $. But 36 is not a perfect cube. So lateral dimensions are irrational.", "But the problem likely expects a clean, exact answer—suggesting the model assumes $ x $ yields rational surface area.", "Wait: perhaps the designer’s ratio and volume were chosen for exactness.", "Let us solve for $ x $ numerically:", "$$\nx^3 = 1080 / 30 = 36 \Rightarrow x = \sqrt[3]{36} \approx 3.3019\n$$", "Then dimensions:", "- $ 2x \approx 6.6038 $\n- $ 3x \approx 9.9057 $\n- $ 5x \approx 16.5107 $", "Surface area:", "$$\nS = 2(6.6038 \cdot 9.9057 + 6.6038 \cdot 16.5107 + 9.9057 \cdot 16.5107)\n$$", "Calculate each:", "- $ 6.6038 \cdot 9.9057 \approx 65.43 $\n- $ 6.6038 \cdot 16.5107 \approx 109.15 $\n- $ 9.9057 \cdot 16.5107 \approx 163.52 $", "Sum: $ 65.43 + 109.15 + 163.52 = 338.10 $", "Double: $ S \approx 676.2 , \ ext{cm}^2 $", "Close to earlier 674.6 — but not perfect.", "Wait—could the dimensions be scaled?", "Suppose the ratio 2:3:5 is correct, volume 1080 cm³.", "Let $ 2x \cdot 3x \cdot 5x = 30x^3 = 1080 \Rightarrow x^3 = 36 \Rightarrow x = \sqrt[3]{36} $", "Then surface area $ S = 62x^2 = 62 \cdot (x^3)^{2/3} = 62 \cdot 36^{2/3} $", "Now $ 36^{2/3} = (6^2)^{2/3} = 6^{4/3} = 6 \cdot 6^{1/3} $", "But $ 6^{1/3} \approx 1.817 $, so $ 6 \cdot 1.817 = 10.902 $, $ 62 \cdot 10.902 \approx 674.76 $", "But perhaps the problem expects exact symbolic form over decimal.", "Alternatively, reconsider: maybe the volume is exactly 1080 and ratio 2:3:5 yields rational $ x $? Only if 30x³ = 1080 → x³ = 36 → x = 36^{1/3}", "But 36^{2/3} = (36^{1/2})^{3/1.5} — no.", "Wait: $ x^3 = 36 \Rightarrow x^2 = 36^{2/3} $", "But $ 36 = 2^2 \cdot 3^2 \Rightarrow 36^{2/3} = (2^2 \cdot 3^2)^{2/3} = 2^{4/3} \cdot 3^{4/3} = (2^4 \cdot 3^4)^{1/3} = (16 \cdot 81)^{1/3} = 1296^{1/3} $", "So $ S = 62 \cdot \sqrt[3]{1296} $", "Factor 1296:", "$ 1296 \div 16 = 81 $, $ 16 = 2^4 $, $ 81 = 3^4 $, so $ 1296 = 2^4 \cdot 3^4 = (2 \cdot 3)^4 \cdot 3^0 = 6^4 $", "Yes! $ 6^4 = 1296 $", "So:", "$$\n\sqrt[3]{1296} = \sqrt[3]{6^4} = 6^{4/3} = 6 \cdot 6^{1/3}\n$$", "So $ S = 62 \cdot 6^{4/3} = 62 \cdot 6 \cdot 6^{1/3} = 372 \cdot \sqrt[3]{6} $", "But is this simplifiable?", "Alternatively, accept decimal?", "But original question likely expects exact simplified expression or clean number.", "Wait—perhaps I miscalculated the volume.", "Let $ l = 2x, w = 3x, h = 5x $, volume $ V = 30x^3 = 1080 \Rightarrow x^3 = 36 \Rightarrow x = \sqrt[3]{36} $", "Surface area $ S = 2(lw + lh + wh) = 2[(2x)(3x) + (2x)(5x) + (3x)(5x)] = 2[6x^2 + 10x^2 + 15x^2] = 2(31x^2) = 62x^2 $", "Now $ x^2 = (x^3)^{2/3} = 36^{2/3} $", "But $ 36^{2/3} = (6^2)^{2/3} = 6^{4/3} $", "But $ 6^{4/3} = (6^1) \cdot (6^{1/3}) = 6 \sqrt[3]{6} $", "So $ S = 62 \cdot 6 \cdot \sqrt[3]{6} = 372 \sqrt[3]{6} $", "This is exact.", "But in SEO context, users want clear, precise, and technically sound content.", "However, perhaps the problem intends for integer dimensions? Let’s reverse: suppose volume 1080, ratio 2:3:5.", "We accept $ x^3 = 36 $, $ x = 36^{1/3} $, so dimensions $ 2\cdot36^{1/3}, 3\cdot36^{1/3}, 5\cdot36^{1/3} $", "Then $ x^2 = 36^{2/3} = (36^{1/3})^2 $", "But $ 36^{2/3} = (6^2)^{2/3} = 6^{4/3} $, same as before.", "Alternatively, the best is to present the exact value.", "But for practical design and marketing, the designer seeks:", "> “A sleek, biodegradable rectangular prism packaging with dimensions in 2:3:5 ratio, volume 1080 cm³, yields a surface area of $ 62 \cdot 36^{2/3} $ cm² — but simplified: since $ 36^{2/3} = (6^2)^{2/3} = 6^{4/3} $, and $ 6^{4/3} = 6 \cdot \sqrt[3]{6} $, so $ S = 62 \cdot 6 \cdot \sqrt[3]{6} = 372 \sqrt[3]{6} $ cm²”", "But this is unwieldy.", "Wait—perhaps the volume is meant to be 1080, but the ratio is set, and designer uses exact math.", "Alternatively, maybe the dimensions are integers and 2:3:5 is approximate? But problem says “in the ratio 2:3:5”.", "Best resolution: compute numerically for clarity and precision.", "From $ x^3 = 36 \Rightarrow x \approx 3.3019 $", "Then:", "- $ l = 6.6038 $\n- $ w = 9.9057 $\n- $ h = 16.5107 $", "Now:", "- $ lw = 6.6038 \ imes 9.9057 \approx 65.43 $\n- $ lh = 6.6038 \ imes 16.5107 \approx 109.15 $\n- $ wh = 9.9057 \ imes 16.5107 \approx 163.52 $\n- Sum = $ 65.43 + 109.15 + 163.52 = 338.10 $\n- Surface area = $ 2 \ imes 338.10 = 676.2 $ cm²", "But 676.2 ≈ $ 372 \sqrt[3]{6} $? Check:", "$ \sqrt[3]{6} \approx 1.817 $, $ 372 \ imes 1.817 \approx 675.4 $ — extremely close.", "So $ S = 372 \sqrt[3]{6} \approx 675.4 $ cm²", "For an exact answer in technical context, use symbolic.", "But for SEO, include clear boxed answer.", "---", "### Final Calculation & Answer", "From volume $ 30x^3 = 1080 \Rightarrow x^3 = 36 \Rightarrow x = \sqrt[3]{36} $", "Dimensions: $ 2x = 2\sqrt[3]{36},\ 3\sqrt[3]{36},\ 5\sqrt[3]{36} $", "Surface area:", "$$\nS = 2( (2x)(3x) + (2x)(5x) + (3x)(5x) ) = 2(6x^2 + 10x^2 + 15x^2) = 2(31x^2) = 62x^2\n$$", "$$\nx^2 = (\sqrt[3]{36})^2 = 36^{2/3} = (6^2)^{2/3} = 6^{4/3} = 6 \cdot 6^{1/3}\n$$", "$$\nS = 62 \cdot 6 \cdot 6^{1/3} = 372 \cdot \sqrt[3]{6}\n$$", "Numerically, $ \sqrt[3]{6} \approx 1.81712 $, so $ S \approx 675.0 , \ ext{cm}^2 $", "But for exactness in engineering and sustainability reporting, the precise form is preferred.", "Thus, the surface area is $ \boxed{372\sqrt[3]{6}} $ cm².", "---", "Optimized SEO Elements:", "- Headline: "Biodegradable Packaging Design: Industrial Designer Creates Efficient 2:3:5 Rectangular Prism Box with Volume 1080 cm³"\n- Key Phrases: “industrial designer biodegradable packaging rectangular prism,” “volume 1080 cm³,” “surface area calculation,” “sustainable industrial design”\n- Benefit: Combines ecological responsibility with precise engineering, ideal for green manufacturing content.\n- Structured Information: Clear problem, derivation, exact answer, real-world application.", "This version balances technical accuracy, SEO strength, and readability for both designers and eco-conscious consumers."]