A manufacturer designs a shipping box in the shape of a cube. The length of each edge of the box...

1. TRANSLATE the problem information

• Given information:
  • A cube-shaped shipping box
  • Each edge length is x inches
  • Need to find volume expression in cubic inches

2. INFER the appropriate formula

• Since we need volume of a cube, we must use the cube volume formula
• Volume of cube = \(\mathrm{(edge\ length)}^3\)
• This makes sense because volume measures 3-dimensional space \(\mathrm{(length} \times \mathrm{width} \times \mathrm{height)}\)

3. Apply the formula

• Substitute the edge length x into the volume formula:
• Volume = \(\mathrm{x}^3\) cubic inches

Answer: D (\(\mathrm{x}^3\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about cube measurements: Students mix up volume with surface area because both involve the number 6 and \(\mathrm{x}^2\).

Surface area of a cube = \(6\mathrm{x}^2\) (6 faces, each with area \(\mathrm{x}^2\)), so students see this familiar pattern and incorrectly select Choice C (\(6\mathrm{x}^2\)) thinking it represents volume.

Second Most Common Error:

Weak INFER skill: Students don't recognize they need the volume formula and instead think about total edge length.

A cube has 12 edges, each of length x, so total edge length = \(12\mathrm{x}\). This leads them to select Choice B (\(12\mathrm{x}\)) even though this measures linear distance, not 3-dimensional volume.

The Bottom Line:

The key challenge is distinguishing between different cube measurements - volume (\(\mathrm{x}^3\)), surface area (\(6\mathrm{x}^2\)), and total edge length (\(12\mathrm{x}\)). Students must recognize that volume specifically requires cubing the edge length because volume measures 3-dimensional space.