A right circular cone has a base circumference of 12pi centimeters and a slant height of 10 centimeters. The volume...

1. TRANSLATE the problem information

• Given information:
  • Right circular cone
  • Base circumference: \(\mathrm{C = 12π\text{ cm}}\)
  • Slant height: \(\mathrm{s = 10\text{ cm}}\)
  • Volume in form: \(\mathrm{V = nπ\text{ cm}^3}\)

• We need to find the value of n

2. INFER what's needed for volume calculation

• Volume formula for cone: \(\mathrm{V = \frac{1}{3}πr^2h}\)
• We need both radius (r) and vertical height (h)
• We have circumference and slant height - these can help us find r and h

3. TRANSLATE circumference to find radius

• Use circumference formula: \(\mathrm{C = 2πr}\)
• Substitute known values: \(\mathrm{12π = 2πr}\)
• SIMPLIFY: Divide both sides by 2π to get \(\mathrm{r = 6\text{ cm}}\)

4. INFER the relationship between slant height, radius, and vertical height

• In a right circular cone, these three measurements form a right triangle
• The vertical height and radius are the legs, slant height is the hypotenuse
• This means we can use Pythagorean theorem: \(\mathrm{s^2 = r^2 + h^2}\)

5. SIMPLIFY to find vertical height

• Substitute known values: \(\mathrm{10^2 = 6^2 + h^2}\)
• Calculate: \(\mathrm{100 = 36 + h^2}\)
• Solve for h²: \(\mathrm{h^2 = 64}\)
• Take square root: \(\mathrm{h = 8\text{ cm}}\)

6. SIMPLIFY to find volume

• Use volume formula: \(\mathrm{V = \frac{1}{3}πr^2h}\)
• Substitute values: \(\mathrm{V = \frac{1}{3}π(6^2)(8)}\)
• Calculate:
  \(\mathrm{V = \frac{1}{3}π(36)(8)}\)
  \(\mathrm{V = \frac{1}{3}π(288)}\)
  \(\mathrm{V = 96π\text{ cm}^3}\)

Answer: B) 96

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to find the vertical height, not use the slant height directly in the volume formula. They might try \(\mathrm{V = \frac{1}{3}πr^2s}\) instead of \(\mathrm{V = \frac{1}{3}πr^2h}\), using slant height (10) instead of vertical height (8).

This leads to \(\mathrm{V = \frac{1}{3}π(36)(10) = 120π}\), causing them to select Choice C (120).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need for Pythagorean theorem but make calculation errors when solving \(\mathrm{10^2 = 6^2 + h^2}\). They might calculate \(\mathrm{100 - 36 = 64}\) incorrectly or make errors taking the square root.

This leads to wrong height values and incorrect final volume calculations, causing confusion and guessing among the remaining answer choices.

The Bottom Line:

This problem tests whether students can distinguish between slant height and vertical height in cone problems, requiring both geometric insight and careful algebraic execution across multiple steps.