The volume, in cubic centimeters, of liquid in a cylindrical can with a fixed radius is given by the function...
GMAT Advanced Math : (Adv_Math) Questions
The volume, in cubic centimeters, of liquid in a cylindrical can with a fixed radius is given by the function V, where \(\mathrm{V(h) = 100πh}\) and h is the height of the liquid in centimeters. Which of the following is the best interpretation of the statement \(\mathrm{V(8) = 800π}\) in this context?
1. TRANSLATE the function notation
- Given information:
- \(\mathrm{V(h) = 100\pi h}\) (volume function)
- \(\mathrm{h}\) = height of liquid in centimeters
- \(\mathrm{V(h)}\) = volume in cubic centimeters
- Statement to interpret: \(\mathrm{V(8) = 800\pi}\)
- In function notation, the number in parentheses is the input, and the number after the equals sign is the output
2. TRANSLATE the specific statement \(\mathrm{V(8) = 800\pi}\)
- \(\mathrm{V(8) = 800\pi}\) means:
- Input: \(\mathrm{h = 8}\) (height is 8 centimeters)
- Output: \(\mathrm{V = 800\pi}\) (volume is \(\mathrm{800\pi}\) cubic centimeters)
3. INFER verification strategy
- To double-check, substitute \(\mathrm{h = 8}\) into the original function:
- \(\mathrm{V(8) = 100\pi(8) = 800\pi}\) ✓
- This confirms our interpretation is correct
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse which number represents the input versus the output in function notation \(\mathrm{V(8) = 800\pi}\).
They might think the \(\mathrm{800\pi}\) represents the input (height) and 8 represents the output (volume), leading them to interpret this as "height of 800 centimeters gives volume of \(\mathrm{8\pi}\) cubic centimeters."
This may lead them to select Choice C (A can with a liquid height of 800 centimeters has a liquid volume of \(\mathrm{8\pi}\) cubic centimeters).
Second Most Common Error:
Inadequate TRANSLATE execution: Students correctly identify that 8 is the input but make an arithmetic error when checking their interpretation.
They might calculate \(\mathrm{V(8) = 100\pi(8) = 64\pi}\) instead of \(\mathrm{800\pi}\), then conclude that the volume should be \(\mathrm{64\pi}\) when height is 8.
This may lead them to select Choice D (A can with a liquid height of 8 centimeters has a liquid volume of \(\mathrm{64\pi}\) cubic centimeters).
The Bottom Line:
Success requires carefully translating function notation - remembering that the value in parentheses is always the input, and the value after the equals sign is always the output. The context (height and volume) doesn't change this fundamental rule of function notation.