The volume of a right circular cone is 48pi cubic centimeters. If the height of the cone is 9 centimeters,...

1. TRANSLATE the problem information

• Given information:
  • Volume = 48π cubic centimeters
  • Height = 9 centimeters
  • Need to find: radius of the base
• This tells us we need to use the cone volume formula to work backwards from volume to radius

2. TRANSLATE to set up the equation

• Use the cone volume formula: \(\mathrm{V = \frac{1}{3}\pi r^2h}\)
• Substitute known values: \(\mathrm{48\pi = \frac{1}{3}\pi r^2(9)}\)
• This gives us an equation we can solve for r

3. SIMPLIFY by eliminating π

• Divide both sides by π: \(\mathrm{48\pi \div \pi = \frac{1}{3}\pi r^2(9) \div \pi}\)
• This simplifies to: \(\mathrm{48 = \frac{1}{3}r^2(9)}\)

4. SIMPLIFY the right side

• Multiply \(\mathrm{\frac{1}{3} \times 9 = 3}\)
• So we have: \(\mathrm{48 = 3r^2}\)

5. SIMPLIFY to isolate r²

• Divide both sides by 3: \(\mathrm{48 \div 3 = 16}\)
• Therefore: \(\mathrm{r^2 = 16}\)

6. SIMPLIFY to find r

• Take the square root of both sides: \(\mathrm{r = \sqrt{16} = 4}\)
• Since radius must be positive, \(\mathrm{r = 4}\) centimeters

Answer: B (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the multi-step algebraic process, particularly when simplifying fractions or performing division.

For example, they might calculate \(\mathrm{48 \div 3 = 15}\) instead of 16, leading to \(\mathrm{r^2 = 15}\) and \(\mathrm{r = \sqrt{15} \approx 3.9}\). This causes confusion since 3.9 isn't among the answer choices, leading to guessing or selecting the closest value.

Second Most Common Error:

Poor TRANSLATE reasoning: Students set up the equation incorrectly, either by forgetting the (1/3) factor in the cone volume formula or by confusing which variable represents what.

Some might use the wrong formula entirely (like cylinder volume \(\mathrm{V = \pi r^2h}\)) or mix up radius and height in their substitution. This leads to completely incorrect equations and wrong final answers, potentially causing them to select Choice D (16) if they mistake \(\mathrm{r^2 = 16}\) as the final answer.

The Bottom Line:

This problem requires careful algebraic manipulation with multiple steps where small arithmetic errors compound. Success depends on methodical simplification while keeping track of what each variable represents throughout the solution process.