As part of a science project on evaporation, Amaya measured the height of a liquid in a container over a...
GMAT Algebra : (Alg) Questions
As part of a science project on evaporation, Amaya measured the height of a liquid in a container over a period of time. The function \(\mathrm{f(x) = 33 - 0.18x}\) gives the estimated height, in centimeters (cm), of the liquid in the container \(\mathrm{x}\) days after the start of the project. Which of the following is the best interpretation of \(\mathrm{33}\) in this context?
The estimated height, in cm, of the liquid at the start of the project
The estimated height, in cm, of the liquid at the end of the project
The estimated change in the height, in cm, of the liquid each day
The estimated number of days for all of the liquid to evaporate
1. TRANSLATE the problem information
- Given function: \(\mathrm{f(x) = 33 - 0.18x}\)
- \(\mathrm{f(x)}\) = estimated height in cm of liquid in container
- \(\mathrm{x}\) = number of days after the start of the project
- Question asks: What does 33 represent?
2. INFER the relationship between function form and context
- This is a linear function in the form \(\mathrm{f(x) = a + bx}\)
- In any linear function, the constant term represents the value when \(\mathrm{x = 0}\)
- Since \(\mathrm{x}\) represents days after the start, \(\mathrm{x = 0}\) means "at the start of the project"
3. TRANSLATE x = 0 to find the initial height
- At the start of the project: \(\mathrm{x = 0}\)
- \(\mathrm{f(0) = 33 - 0.18(0)}\)
\(\mathrm{= 33 - 0}\)
\(\mathrm{= 33}\) - Therefore, 33 represents the height of liquid at the start of the project
Answer: A. The estimated height, in cm, of the liquid at the start of the project
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may not connect "x days after the start" with the idea that \(\mathrm{x = 0}\) represents the starting point. They might think 33 is just "some number in the equation" without understanding its contextual meaning.
This leads to confusion about what each part of the function represents, causing them to guess among the answer choices.
Second Most Common Error:
Poor INFER reasoning: Students might recognize that 33 is important but confuse it with other function characteristics. They may think 33 represents the rate of change (which is actually \(\mathrm{-0.18}\)) or the final value.
This may lead them to select Choice C (\(\mathrm{-0.18}\) is the actual rate of change) or make other incorrect connections.
The Bottom Line:
Success on this problem requires understanding both the mathematical structure of linear functions and how to interpret function components within a real-world context. The key insight is recognizing that the constant term always represents the starting value when time = 0.
The estimated height, in cm, of the liquid at the start of the project
The estimated height, in cm, of the liquid at the end of the project
The estimated change in the height, in cm, of the liquid each day
The estimated number of days for all of the liquid to evaporate