Cube A has edge lengths that are 12 times the edge lengths of cube B. The volume of cube A...

1. TRANSLATE the problem information

• Given information:
  • Cube A's edge lengths are 12 times cube B's edge lengths
  • Volume of cube A is k times volume of cube B
  • Need to find: value of k

2. TRANSLATE to mathematical relationships

• Let cube B have edge length \(\mathrm{s}\)
• Then cube A has edge length \(\mathrm{12s}\)
• Volume relationship: \(\mathrm{Volume_A = k \times Volume_B}\)

3. INFER the solution approach

• To find k, I need to express both volumes in terms of \(\mathrm{s}\)
• Then I can set up the ratio and solve for k
• Key insight: Volume changes by the cube of the linear scale factor

4. SIMPLIFY using volume formula

• Volume of cube B = \(\mathrm{s^3}\)
• Volume of cube A = \(\mathrm{(12s)^3 = 12^3 \times s^3 = 1728s^3}\)

5. SIMPLIFY to find k

• Since \(\mathrm{Volume_A = k \times Volume_B}\):
• \(\mathrm{1728s^3 = k \times s^3}\)
• \(\mathrm{k = 1728}\)

Answer: 1728

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that volume scaling requires cubing the linear scale factor.

Many students think: "If the edge is 12 times bigger, then the volume is also 12 times bigger." They forget that volume is a three-dimensional measure, so the scale factor must be cubed. This leads them to incorrectly answer \(\mathrm{k = 12}\) instead of \(\mathrm{k = 1728}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the problem correctly but make arithmetic errors.

They correctly identify that \(\mathrm{k = 12^3}\) but then calculate \(\mathrm{12^3}\) incorrectly. Common wrong calculations include \(\mathrm{12^3 = 144}\) (confusing with \(\mathrm{12^2}\)) or other computational mistakes. This leads to various incorrect numerical answers.

The Bottom Line:

This problem tests understanding of how linear scaling affects volume in three-dimensional objects. The key insight is that when linear dimensions scale by factor \(\mathrm{n}\), volume scales by factor \(\mathrm{n^3}\).