A right circular cylinder has a volume of 45pi. If the height of the cylinder is 5, what is the...

1. TRANSLATE the problem information

• Given information:
  • Volume of right circular cylinder = \(45\pi\)
  • Height = \(5\)
  • Need to find: radius

• What this tells us: We can use the volume formula with known values to solve for the unknown radius.

2. INFER the approach

• We need the volume formula for a cylinder: \(\mathrm{V} = \pi r^2h\)
• Strategy: Substitute known values and solve for r
• This becomes an algebraic equation we can solve step-by-step

3. TRANSLATE the setup into an equation

Set up: \(45\pi = \pi r^2(5)\)

This simplifies to: \(45\pi = 5\pi r^2\)

4. SIMPLIFY through algebraic steps

• Divide both sides by \(5\pi\):
  \(45\pi \div 5\pi = r^2\)
  \(9 = r^2\)

• Take the square root of both sides:
  \(r = \pm\sqrt{9} = \pm 3\)

5. APPLY CONSTRAINTS to select final answer

• Since radius must be positive in real-world context:
  \(r = 3\)

Answer: A. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: After finding that \(r^2 = 9\), students might divide 9 by 2 (getting 4.5) instead of taking the square root. This error occurs when students confuse the operation needed to "undo" squaring.

This may lead them to select Choice B (4.5).

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly find that \(r^2 = 9\) but then think they're done, reporting the value of \(r^2\) as the radius rather than taking the final step of finding r itself.

This may lead them to select Choice C (9).

The Bottom Line:

This problem tests whether students can correctly reverse the squaring operation. The key insight is recognizing that when you have \(r^2 = 9\), you need \(\sqrt{9} = 3\), not \(9 \div 2 = 4.5\). Many students struggle with this because they're not confident about which operation undoes squaring.