A cylindrical tank has a height that is 4 times its radius. The function \(\mathrm{V(r) = 4\pi r^3}\) gives the...

1. TRANSLATE the function notation

• Given information:
  • \(\mathrm{V(r)} = 4\pi\mathrm{r}^3\) represents volume when radius is r meters
  • We need to interpret \(\mathrm{V(3)} = 108\pi\)
• What this tells us: \(\mathrm{V(3)}\) means "substitute \(\mathrm{r} = 3\) into the function"

2. INFER what the equation represents

• In function notation, \(\mathrm{V(3)} = 108\pi\) means:
  • Input: radius = 3 meters
  • Output: volume = \(108\pi\) cubic meters
• This is a cause-and-effect relationship: "If radius is 3, then volume is \(108\pi\)"

3. TRANSLATE to verify our interpretation

• Let's check:
  \(\mathrm{V(3)} = 4\pi(3)^3\)
  \(= 4\pi(27)\)
  \(= 108\pi\) ✓
• This confirms our interpretation is correct

4. INFER which answer choice matches

• We need: "If radius is 3 meters, then volume is \(108\pi\) cubic meters"
• Choice (A) states exactly this relationship

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students confuse what the input and output represent in function notation.

They might think \(\mathrm{V(3)}\) means "when the volume is 3" instead of "when the radius is 3," or they might think the output \(108\pi\) represents height instead of volume. This confusion about which variable goes where leads them to select Choice (C) (using height as input) or Choice (B) (interpreting volume as height).

Second Most Common Error:

Poor INFER reasoning: Students reverse the cause-and-effect relationship in the function.

They might read \(\mathrm{V(3)} = 108\pi\) as "\(108\pi\) causes 3" instead of "3 causes \(108\pi\)," leading them to think \(108\pi\) is the input value. This backwards thinking may lead them to select Choice (D) (treating \(108\pi\) as the height input).

The Bottom Line:

Function notation requires careful attention to what goes in (input) versus what comes out (output), and students must distinguish between the different measurements (radius, height, volume) in the context.