Cubes PQRS and TUVW are similar. The length of each edge of TUVW is 3 times the length of the...

1. TRANSLATE the problem information

• Given information:
  • Cubes PQRS and TUVW are similar
  • Each edge of TUVW is \(3\) times the corresponding edge of PQRS
  • Volume of PQRS = \(64\) cubic units
• What this tells us: We have a linear scale factor of \(3\) between the similar cubes

2. INFER the scaling relationship

• For similar three-dimensional figures, volumes don't scale by the same factor as linear dimensions
• Key insight: When linear dimensions scale by factor k, volumes scale by \(\mathrm{k}^3\)
• Since our linear scale factor is \(3\), our volume scale factor is \(3^3\)

3. SIMPLIFY the volume scale factor calculation

• Calculate the volume scale factor: \(3^3 = 3 \times 3 \times 3 = 27\)
• Volume of TUVW = Volume of PQRS \(\times\) \(27\) = \(64 \times 27 = 1,728\)

Answer: C (1,728)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students assume volume scales by the same factor as linear dimensions (\(3\) instead of \(3^3\))

They think: "If each edge is \(3\) times longer, then the volume is also \(3\) times larger."
This leads them to calculate: \(64 \times 3 = 192\)

This may lead them to select Choice A (192)

Second Most Common Error:

Conceptual confusion about scaling: Students remember that area scales by \(\mathrm{k}^2\), so they incorrectly apply this to volume

They think: "I remember something about squaring the scale factor, so volume scales by \(3^2 = 9\)."
This leads them to calculate: \(64 \times 9 = 576\)

This may lead them to select Choice B (576)

The Bottom Line:

The key challenge is remembering that three-dimensional volume scales by the cube of the linear scale factor, not by the linear scale factor itself. Students who don't have this scaling relationship firmly established will struggle to get beyond the setup phase of this problem.