A right circular cone has a volume of 71,148pi cubic centimeters and the area of its base is 5,929pi square...

1. TRANSLATE the problem information

• Given information:
  • Volume = \(\mathrm{71{,}148\pi}\) cubic centimeters
  • Base area = \(\mathrm{5{,}929\pi}\) square centimeters
• Find: slant height in centimeters

2. INFER the solution strategy

• Key insight: Slant height requires knowing both radius and height of the cone
• We have volume and base area, but not height or radius directly
• Strategy: Use volume formula to find height, base area to find radius, then Pythagorean theorem for slant height

3. SIMPLIFY to find the height

• Use cone volume formula: \(\mathrm{V = \frac{1}{3}\pi r^2h}\)
• Since base area = \(\mathrm{\pi r^2}\), we can substitute:
  \(\mathrm{71{,}148\pi = \frac{1}{3}(5{,}929\pi)(h)}\)
• Divide both sides by \(\mathrm{5{,}929\pi}\):
  \(\mathrm{12 = \frac{1}{3}h}\)
• Multiply by 3: \(\mathrm{h = 36\text{ cm}}\)

4. SIMPLIFY to find the radius

• From base area formula: \(\mathrm{\pi r^2 = 5{,}929\pi}\)
• Divide by \(\mathrm{\pi}\): \(\mathrm{r^2 = 5{,}929}\)
• Take square root: \(\mathrm{r = \sqrt{5{,}929} = 77\text{ cm}}\) (use calculator)

5. INFER the geometric relationship and apply Pythagorean theorem

• In a cone, the radius, height, and slant height form a right triangle
• Radius and height are the legs; slant height is the hypotenuse
• SIMPLIFY: \(\mathrm{s^2 = r^2 + h^2 = 5{,}929 + 36^2 = 5{,}929 + 1{,}296 = 7{,}225}\)
• Take square root: \(\mathrm{s = \sqrt{7{,}225} = 85\text{ cm}}\) (use calculator)

Answer: D. 85

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that slant height requires finding both radius and height first. They may try to work directly with the given volume and base area, looking for a formula that directly connects these to slant height. Since no such direct relationship exists, this leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the multi-step approach but make calculation errors, particularly when finding the height (getting confused about the 1/3 factor) or when computing the final square roots. A common mistake is calculating \(\mathrm{36^2}\) incorrectly or making arithmetic errors when adding \(\mathrm{5{,}929 + 1{,}296}\). This may lead them to select Choice A (12) if they mistake the height for the slant height, or other incorrect choices based on their calculation errors.

The Bottom Line:

This problem requires recognizing that the slant height depends on the cone's internal geometry (the right triangle formed by radius, height, and slant height), not just the given measurements. Success depends on systematically working through multiple formulas in the correct sequence.