Cube A has an edge length of s centimeters. Cube B has a total sum of all its edge lengths...

1. INFER what 'sum of all edge lengths' means

• Given information:
  • Cube A has edge length \(\mathrm{s}\)
  • Cube B's total edge sum is 84 more than Cube A's total edge sum
• Key insight: A cube has 12 edges, so 'sum of all edge lengths' = \(\mathrm{12 \times (individual\ edge\ length)}\)

2. TRANSLATE the relationship into equations

• Cube A's total edge sum = \(\mathrm{12s}\)
• Cube B's total edge sum = \(\mathrm{12s + 84}\) (84 greater than Cube A)

3. INFER how to find Cube B's individual edge length

• If Cube B has edge length \(\mathrm{e_B}\), then: \(\mathrm{12 \times e_B = 12s + 84}\)
• Strategy: Solve for \(\mathrm{e_B}\) to get the individual edge length

4. SIMPLIFY the equation algebraically

• \(\mathrm{12 \times e_B = 12s + 84}\)
• Divide both sides by 12: \(\mathrm{e_B = \frac{12s + 84}{12}}\)
• SIMPLIFY: \(\mathrm{e_B = s + 7}\)

5. INFER the volume function

• Volume of any cube = \(\mathrm{(edge\ length)^3}\)
• Volume of Cube B = \(\mathrm{(e_B)^3 = (s + 7)^3}\)
• Therefore: \(\mathrm{g(s) = (s + 7)^3}\)

Answer: A. \(\mathrm{g(s) = (s + 7)^3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that a cube has 12 edges, so they think 'sum of all edge lengths' just equals the individual edge length.

They might think: 'If Cube A has edge \(\mathrm{s}\), and Cube B's edge sum is 84 more, then Cube B has edge length \(\mathrm{s + 84}\).'

This leads them to calculate volume as \(\mathrm{(s + 84)^3}\) and select Choice C.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{12 \times e_B = 12s + 84}\) but make algebraic mistakes when solving.

They might divide incorrectly and get \(\mathrm{e_B = s + 12}\) or \(\mathrm{e_B = s + 14}\), leading them to select Choice D or Choice B respectively.

The Bottom Line:

This problem tests whether students understand the geometric structure of a cube (12 edges) and can properly handle the relationship between individual measurements and totals. The key breakthrough is realizing that 'sum of all edge lengths' requires multiplying the individual edge length by 12.