A cylinder has a diameter of 8 inches and a height of 12 inches. What is the volume, in cubic...

Step-by-Step Solution

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Diameter = 8\text{ inches}}\)
  • \(\mathrm{Height = 12\text{ inches}}\)
• We need to find the volume in cubic inches

2. INFER the approach and what we need

• To find cylinder volume, we need the formula \(\mathrm{V = πr^2h}\)
• The formula requires radius, but we're given diameter
• Since \(\mathrm{radius = diameter ÷ 2}\), we have: \(\mathrm{r = 8 ÷ 2 = 4\text{ inches}}\)

3. INFER which formula to apply

• Use the cylinder volume formula: \(\mathrm{V = πr^2h}\)
• We have: \(\mathrm{r = 4\text{ inches, }h = 12\text{ inches}}\)

4. SIMPLIFY the calculation

• \(\mathrm{V = π(4)^2(12)}\)
• \(\mathrm{V = π(16)(12)}\)
• \(\mathrm{V = 192π\text{ cubic inches}}\)

Answer: C. 192π

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about radius vs diameter: Students might use the diameter (8) directly in place of radius in the formula \(\mathrm{V = πr^2h}\).

This gives them \(\mathrm{V = π(8)^2(12) = π(64)(12) = 768π}\), leading them to select Choice D (768π).

Second Most Common Error:

Weak INFER skill: Students correctly find the area of the circular base (\(\mathrm{πr^2 = π(4)^2 = 16π}\)) but forget that volume requires multiplying by height.

They stop at the area calculation and select Choice A (16π) instead of continuing to multiply by the height.

The Bottom Line:

This problem tests whether students can distinguish between diameter and radius, and whether they understand that cylinder volume requires both the base area AND the height. The key insight is remembering that the volume formula needs radius, not diameter, and that finding volume is a two-part process: base area times height.