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
What is included
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.
Skills assessed
- Constrained one-variable optimization
Prerequisites
- derivative rules
- critical points
Long description
Read the diagram from top to bottom. Each numbered box names one decision or mathematical operation. Arrows show the required order; the text labels remain the complete interpretation in print, dark mode, and nonvisual reading.
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- Squares of side x are cut from a 12 by 12 sheet. Maximize the open box volume.
Complete worked solutions
Every problem has a source-matched answer and independently reviewed derivation.
01
Problem 1: Squares of side x are cut from a 12 by 12 sheet. Maximize the open box volume.
Answer:
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.
Common errors
- Solving f'(x)=0 without building the constraint or checking the feasible domain.