A rectangular box with a square base is to be made to hold 108 cubic meters of volume. The cost is $5 per square meter for the base and $3 per square meter for each of the four sides. What is the minimum cost?
["What’s Hidden in the Stack of Steel and Wood? The Cost of Building a 108-Meter-Volume Box on a Budget", "Have you ever wondered how engineers and designers balance shape, strength, and cost when creating structures that hold exactly 108 cubic meters? The answer lies in a simple yet powerful design: a rectangular box with a square base. In the U.S. market, where efficiency meets estimation in construction and manufacturing, this shape commonly arises in everything from storage units to modular display systems. Now, as costs rise across materials, understanding how to minimize them using smart geometry becomes crucial—especially when the volume is fixed at 108 cubic meters and pricing varies dramatically per surface.", "This exact configuration—base forming a perfect square and four identical rectangular sides—optimizes space while simplifying manufacturing. But pricing varies: $5 per square meter for the base and $3 per square meter for each vertical side. The key question on minds rounding this calculation: what’s the absolute minimum cost, and how do pricing dynamics shape that outcome? This isn’t just math—it’s a real-world problem driving smart budgeting in industry, design, and planning.", "---", "### The Math Behind the Minimum Cost", "A rectangular box with a square base has dimensions defined by a square base of side s meters and height h meters. Given the volume must be exactly 108 cubic meters, the math equation starts here: \n\( s^2 \cdot h = 108 \)", "From here, we express height in terms of base size: \n\( h = \frac{108}{s^2} \)", "Surface area and cost depend on: \n- Base: 1 piece at $5 per sq m → cost: \( 5s^2 \) \n- Four sides: each side area \( s \ imes h \), total cost: \( 4 \cdot 3 \cdot s \cdot h = 12sh \)", "Substitute height into cost expression: \nTotal cost = \( 5s^2 + 12s \cdot \frac{108}{s^2} \) \nSimplify: \n\("]
