Cheapest open-top box

MediumCalculus~4m

An open-top box has a square base and must hold a volume of $32000$ cm³. Material is used only for the base and the four sides (no top). What is the minimum total surface area of material (in cm²)?

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No top: $S = x^2 + 4xh$ , with $x^2 h = 32000$ .
Reduce to $S(x) = x^2 + \tfrac{128000}{x}$ and minimise.

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4800

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Cheapest open-top box

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