A right circular cone has a volume of 1/3pi cubic feet and a height of 9 feet. What is the...

1. TRANSLATE the problem information

• Given information:
  • Right circular cone
  • Volume = \(\frac{1}{3}\pi\) cubic feet
  • Height = \(9\) feet
  • Need to find: radius

• What this tells us: We need to use the cone volume formula with known V and h to find r

2. INFER the approach

• We have volume and height, need radius
• The cone volume formula \(\mathrm{V} = \frac{1}{3}\pi r^2h\) connects all three quantities
• Strategy: substitute known values and solve for r

3. TRANSLATE into mathematical equation

Set up: \(\frac{1}{3}\pi = \frac{1}{3}\pi r^2(9)\)

4. SIMPLIFY through algebraic steps

• Divide both sides by \(\frac{1}{3}\pi\):
  \(1 = 9r^2\)

• Divide both sides by 9:
  \(r^2 = \frac{1}{9}\)

• Take the square root:
  \(r = \sqrt{\frac{1}{9}} = \frac{1}{3}\)

5. APPLY CONSTRAINTS to select final answer

• Since radius must be positive in real-world context:
  \(r = \frac{1}{3}\) feet

Answer: A. \(\frac{1}{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when manipulating the equation or evaluating the square root.

For example, they might incorrectly simplify \(\sqrt{\frac{1}{9}}\) as \(\frac{\sqrt{3}}{3}\) or \(\frac{1}{\sqrt{3}}\), or make errors in the fraction arithmetic when dividing by \(\frac{1}{3}\pi\). These calculation mistakes can lead them to select Choice B (\(\frac{1}{\sqrt{3}}\)) or Choice C (\(\sqrt{3}\)).

Second Most Common Error:

Poor algebraic setup: Students might incorrectly set up or solve the equation, perhaps confusing which variable to solve for or making errors that lead to \(r^2 = 9\) instead of \(r^2 = \frac{1}{9}\).

This fundamental error in equation solving would give them \(r = 3\), leading them to select Choice D (\(3\)).

The Bottom Line:

This problem requires careful algebraic manipulation with fractions and radicals. Success depends on systematically working through each step while maintaining precision with fraction operations and square root evaluation.