Find a formula for the described function and state its...

Find a formula for the described function and state its domain.
An open rectangular box with volume $2 \mathrm{~m}^3$ has a square base. Express the surface area of the box as a function of the length of a side of the base.

Key Concepts

Surface Area Calculation

Calculating the surface area requires summing the areas of each of the box's faces while accounting for the box being open (i.e., missing the top). For a box with a square base, this typically involves adding the area of the base to the area of the four side faces. When the height is expressed in terms of the side length via the volume constraint, the entire surface area can be written as a function of a single variable, which simplifies analysis and optimization tasks.

Domain Analysis

After obtaining an expression for the surface area as a function of the base's side length, it is important to determine the domain of the function. This involves identifying all valid values for the side length that result in a physically realistic box. Since dimensions must be positive and the height determined by the volume constraint must also be positive, the domain is restricted to positive values that yield a meaningful, non-degenerate geometric shape.

Geometric Modeling

This concept involves representing a physical object—in this case, an open rectangular box with a square base—using mathematical equations. It requires identifying the dimensions of the object and formulating relationships between them, such as establishing variables for the side of the square base and the height of the box. This modeling is essential for translating geometric properties into algebraic expressions that describe quantities like volume and surface area.

Volume Constraint

The volume constraint provides a critical relationship between the dimensions of the box. Given that the volume is fixed, one can express the height of the box in terms of the side length of the base. Specifically, if the base side length is designated as x and the box’s volume is fixed, the height h can be derived by rearranging the volume formula. This substitution simplifies other calculations and embeds the physical constraint directly into further expressions.

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