The surface area of a cube is \(6(\mathrm{a}/4)^2\), where a is a positive constant. Which of the following gives the...

1. TRANSLATE the problem information

• Given information:
  • Total surface area of cube = \(6(\mathrm{a}/4)^2\)
  • \(\mathrm{a}\) is a positive constant
  • Need perimeter of one face

2. INFER the connection between total and individual face area

• Key insight: A cube has 6 identical square faces
• Since total surface area = \(6 \times \text{(area of one face)}\):
  • Area of one face = \(6(\mathrm{a}/4)^2 \div 6 = (\mathrm{a}/4)^2\)

3. INFER how to find side length from face area

• Each face is a square with area \((\mathrm{a}/4)^2\)
• Since \(\text{area} = \text{side}^2\), we need: \(\text{side} = \sqrt{\text{area}}\)
• Side length = \(\sqrt{(\mathrm{a}/4)^2} = \mathrm{a}/4\)

4. SIMPLIFY to find the perimeter

• Perimeter of square face = \(4 \times \text{side length}\)
• Perimeter = \(4 \times (\mathrm{a}/4) = \mathrm{a}\)

Answer: B. \(\mathrm{a}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to work backwards from total surface area to individual face properties. They might try to work directly with the given expression \(6(\mathrm{a}/4)^2\) without understanding what it represents or how to break it down to find properties of a single face.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need the side length \(\mathrm{a}/4\), but make algebraic errors when calculating the perimeter. They might incorrectly compute \(4 \times (\mathrm{a}/4)\) as \(\mathrm{a}/4\) instead of \(\mathrm{a}\), or make errors with the parentheses.

This may lead them to select Choice A (\(\mathrm{a}/4\)).

The Bottom Line:

This problem tests whether students can systematically deconstruct a cube's total surface area to find properties of individual faces. The key challenge is recognizing that you must work backwards through the geometric relationships rather than trying to manipulate the given expression directly.