A concrete cylindrical column has a diameter of 12 inches and a height of 12 inches. What is the volume...

1. TRANSLATE the problem information

• Given information:
  • Diameter of cylindrical column = 12 inches
  • Height of cylindrical column = 12 inches
• Need to find: Volume in cubic inches

2. INFER what formula and values to use

• Volume of a cylinder requires the formula \(\mathrm{V = \pi r^2h}\)
• The formula needs radius (r), not diameter
• Need to find radius first: \(\mathrm{radius = diameter \div 2}\)

3. SIMPLIFY to find the radius

• \(\mathrm{radius = 12 \div 2 = 6}\) inches

4. SIMPLIFY by applying the volume formula

• \(\mathrm{V = \pi r^2h}\)
• \(\mathrm{V = \pi(6^2)(12)}\)
• \(\mathrm{V = \pi(36)(12)}\)
• \(\mathrm{V = 432\pi}\) cubic inches (use calculator if needed for \(\mathrm{36 \times 12}\))

Answer: B. \(\mathrm{432\pi}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing INFER step: Students might try to use the diameter directly in the volume formula instead of converting to radius first.

If they substitute diameter = 12 directly:

\(\mathrm{V = \pi(12^2)(12)}\)

\(\mathrm{= \pi(144)(12)}\)

\(\mathrm{= 1728\pi}\)

Since \(\mathrm{1728\pi}\) isn't among the answer choices, this leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating \(\mathrm{6^2 \times 12}\).

For example, they might calculate \(\mathrm{6^2 = 36}\) correctly but then compute \(\mathrm{36 \times 12}\) incorrectly, potentially getting values that lead them toward wrong answer choices through guessing.

The Bottom Line:

The key challenge is remembering that cylinder volume formulas use radius, not diameter. Students must pause and INFER this relationship rather than rushing directly into formula substitution.