Find the dimensions giving the minimum surface area, given...

Find the dimensions giving the minimum surface area, given that the volume is $8 \mathrm{cm}^{3}$.
A closed rectangular box, with a square base $x$ by $x \mathrm{cm}$ and height $h \space \mathrm{cm} .$

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Key Concepts

Substitution Method

The substitution method involves using the constraint (in this case, the fixed volume) to express one variable in terms of another. This reduces the number of variables in the optimization problem, simplifying the process of finding the derivative and solving for the optimal dimensions.

Differential Calculus

Differential calculus is used to find the minimum or maximum of a function by taking its derivative and finding the critical points. Here, differentiating the surface area function with respect to one of the dimensions after applying the constraint helps identify the dimensions that minimize the surface area.

Constrained Optimization

Constrained optimization involves finding the optimal (minimum or maximum) value of a function subject to a constraint. In this problem, it means minimizing the surface area while keeping the volume fixed, which is a common scenario in real-world design problems.

Surface Area and Volume Relationships

Understanding the mathematical relationships between surface area and volume is key. For a box with a square base, knowing the formulas for these quantities allows one to express the surface area in terms of a single variable by incorporating the volume constraint.

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