For a cylinder with positive radius r and height h, the volume V is given by the formula V =...

1. TRANSLATE the problem information

• Given formula: \(\mathrm{V = \pi r^2h}\)
• Goal: Express h in terms of V and r

2. INFER the algebraic strategy

• Notice that h is being multiplied by \(\mathrm{\pi r^2}\)
• To isolate h, I need to "undo" this multiplication by dividing both sides by \(\mathrm{\pi r^2}\)
• This will leave h by itself on one side

3. SIMPLIFY by applying division to both sides

• Divide both sides by \(\mathrm{\pi r^2}\):
  \(\mathrm{V \div (\pi r^2) = (\pi r^2h) \div (\pi r^2)}\)

• On the right side, the \(\mathrm{\pi r^2}\) terms cancel:
  \(\mathrm{V/(\pi r^2) = h}\)

• Rewrite with h on the left:
  \(\mathrm{h = V/(\pi r^2)}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may think they should subtract \(\mathrm{\pi r^2}\) from both sides instead of dividing by it.

When students see \(\mathrm{V = \pi r^2h}\), they might think "to get h alone, I need to get rid of \(\mathrm{\pi r^2}\)" and incorrectly reason that subtracting will accomplish this. This leads to \(\mathrm{V - \pi r^2 = h}\), which would make them select Choice A (\(\mathrm{h = V - \pi r^2}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need to divide by \(\mathrm{\pi r^2}\), but make errors in handling the π symbol during algebraic manipulation.

Some students might incorrectly move π to the numerator or forget that r needs to be squared in the denominator. This confusion about where π belongs could lead them to select Choice C (\(\mathrm{h = V\pi/r^2}\)) or Choice D (\(\mathrm{h = V/\pi r}\)).

The Bottom Line:

This problem tests whether students truly understand what "solving for a variable" means - you must undo operations in reverse order, and multiplication is undone by division, not subtraction.