Worked problem · Calculus I · Unit 2B

Open Top Box Optimization: Worked Example and Solution

A complete constrained one-variable optimization example with method choice, derivation, verification, and a common wrong approach.

1 problem8 min estimated timeIntermediate to advanced progression

Problem

Every problem has a source-matched answer and independently reviewed derivation.

Answer: $x=2,\ V_{\max}=128$

Why this method: Constrained one-variable optimization matches the mathematical structure before any algebraic cleanup.

V=x(12-2x)^2 on 0<x<6; V'=12(x-2)(x-6), so the interior maximum is x=2.
Check feasibility, endpoints, and the sign or second derivative before declaring the optimum.
Therefore the result is $x=2,\ V_{\max}=128$ .

See a concise answer and full derivation for squares of side x are cut from a 12 by 12 sheet. maximize the open box volume.

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Solving f'(x)=0 without building the constraint or checking the feasible domain.