A right circular cone has volume 48pi cubic centimeters and height 12 centimeters. What is the radius of the base...

1. TRANSLATE the problem information

• Given information:
  • Volume = \(48\pi\) cubic centimeters
  • Height = \(12\) centimeters
  • Need to find: radius of the base

2. INFER the approach

• Since we have volume and height but need radius, we should use the cone volume formula
• Strategy: Use \(\mathrm{V} = \frac{1}{3}\pi\mathrm{r}^2\mathrm{h}\) and solve for r

3. Set up the equation and SIMPLIFY

• Start with: \(\mathrm{V} = \frac{1}{3}\pi\mathrm{r}^2\mathrm{h}\)
• Substitute known values: \(48\pi = \frac{1}{3}\pi(\mathrm{r}^2)(12)\)
• SIMPLIFY the right side: \(48\pi = 4\pi\mathrm{r}^2\)
• Divide both sides by \(4\pi\): \(\mathrm{r}^2 = 12\)

4. SIMPLIFY to find the final answer

• Take the square root: \(\mathrm{r} = \sqrt{12}\)
• SIMPLIFY the radical: \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\)

Answer: D. \(2\sqrt{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students struggle with radical simplification and incorrectly handle \(\sqrt{12}\).

Many students might leave \(\sqrt{12}\) as their final answer or incorrectly simplify it. For instance, they might think \(\sqrt{12} = 4\sqrt{3}\) (confusing multiplication rules) or simply not recognize that \(\sqrt{12}\) can be simplified. Since \(\sqrt{12}\) isn't among the answer choices, this confusion forces them to guess or abandon their systematic approach.

This leads to confusion and guessing among the given choices.

Second Most Common Error:

Missing conceptual knowledge: Not remembering or incorrectly recalling the cone volume formula.

Students might confuse the cone formula \(\mathrm{V} = \frac{1}{3}\pi\mathrm{r}^2\mathrm{h}\) with other volume formulas (like cylinder: \(\mathrm{V} = \pi\mathrm{r}^2\mathrm{h}\)). If they use the wrong formula, their algebraic work will be correct but lead to an incorrect radius value that matches one of the wrong answer choices.

This may lead them to select Choice A (2) if they accidentally used the cylinder formula.

The Bottom Line:

This problem tests both formula knowledge and radical simplification skills. Success requires knowing the correct cone volume formula AND being able to simplify \(\sqrt{12} = 2\sqrt{3}\). Many students can handle the algebra but stumble on the final radical simplification step.