The volume of a cube is 27 cubic centimeters. What is the length, in centimeters, of one edge of the...

1. TRANSLATE the problem information

• Given information:
  • Volume of cube = 27 cubic centimeters
  • Need to find: length of one edge

2. INFER the solution approach

• To find edge length from volume, I need to "undo" the cubing process
• Since volume involves cubing the edge (\(\mathrm{V = e^3}\)), I need to take the cube root
• Strategy: Use the cube volume formula, then solve for edge length

3. TRANSLATE into mathematical equation

• Volume formula: \(\mathrm{V = e^3}\)
• Substitute known volume: \(\mathrm{e^3 = 27}\)

4. SIMPLIFY to find the edge length

• Take cube root of both sides: \(\mathrm{e = \sqrt[3]{27}}\)
• Calculate: \(\mathrm{\sqrt[3]{27} = 3}\) (since \(\mathrm{3 \times 3 \times 3 = 27}\))

Answer: A. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that cube root is needed to "reverse" the cubing operation. Instead, they might try division operations like \(\mathrm{27 \div 6 = 4.5}\) (incorrectly thinking about 6 faces of a cube), leading to confusion and guessing between the available choices.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember or recognize that \(\mathrm{\sqrt[3]{27} = 3}\). They may set up \(\mathrm{e^3 = 27}\) correctly but then struggle with the cube root calculation. This may lead them to select Choice (D) 27 by incorrectly thinking the edge equals the volume, or Choice (C) 9 by confusing cube and square relationships.

The Bottom Line:

This problem tests whether students can work backwards from a cubed result. The key insight is recognizing that volume involves cubing the edge length, so finding the edge requires the inverse operation - taking the cube root.