Problem

A closed cylinder has radius \(r\), 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 cylinder with radius r, height h, two circular bases, lateral surface, and fixed surface area S; the eliminated model, interval, optimizer, dimensions, and maximum are withheld.

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Use the correct formula but include zero radius or zero height and choose a degenerate zero-volume cylinder as the maximum.

Leave r and h independent and conclude that no one-variable model or optimizer is determined, despite the fixed surface-area constraint.

From 2pi r² + 2pi rh = S, h = (S - 2pi r²)/(2pi r) and V(r) = r(S - 2pi r²)/2, with 0 < r < sqrt(S/(2pi)). The maximum occurs at r = sqrt(S/(6pi)), where h = 2r and Vmax = 2pi(S/(6pi))³/².

Use pi r² + 2pi rh = S, omitting one circular base, and optimize that incomplete surface-area model.

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A closed cylinder has radius \(r\), 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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