Problem

A closed box has square base side \(x\), height \(h\), and fixed surface area \(S~>~0\). Use the diagram and constraint to build a \(\text{one}\text{-}\text{variable}\) volume model with its positive interval, then find the optimizer, dimensions, and maximum volume.

Prompt closed square-base box with base side x, height h, two square bases, four side faces, and fixed surface area S; the eliminated model, feasible interval, optimizer, dimensions, and maximum are withheld.

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Use 2x² + 2xh = S, omitting two side faces, and optimize the resulting incomplete surface-area model.

From 2x² + 4xh = S, h = (S - 2x²)/(4x) and V(x) = x(S - 2x²)/4, with 0 < x < sqrt(S/2). The maximum occurs at x = sqrt(S/6), where h = x and Vmax = (S/6)³/².

Treat the base as independent length l and width w, leaving a two-variable model rather than using the stated square-base side x.

Use the correct formula but include zero-height or zero-base endpoints and choose a degenerate zero-volume box as the maximum.

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A closed box has square base side \(x\), height \(h\), and fixed surface area \(S~>~0\). Use the diagram and constraint to build a \(\text{one}\text{-}\text{variable}\) volume model with its positive interval, then find the optimizer, dimensions, and maximum volume.

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