The function \(\mathrm{P(t) = 24.8(1.036)^t}\) gives the predicted population, in millions, of a certain country for the period from 1984...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{P(t) = 24.8(1.036)^t}\) gives the predicted population, in millions, of a certain country for the period from 1984 to 2018, where t is the number of years after 1984. According to the model, what is the best interpretation of the statement '\(\mathrm{P(8) \approx 32.9}\)'?
In \(1984\), the predicted population of this country was approximately \(8\) million.
In \(1984\), the predicted population of this country was approximately \(32.91\) million.
\(8\) years after \(1984\), the predicted population of this country was approximately \(32.91\) million.
\(32.91\) years after \(1984\), the predicted population of this country was approximately \(8\) million.
1. TRANSLATE the function notation
- Given: \(\mathrm{P(t) = 24.8(1.036)^t}\) where \(\mathrm{t}\) is years after 1984
- Statement: "\(\mathrm{P(8)}\) is approximately equal to 32.9"
- This means: When \(\mathrm{t = 8}\), the function value \(\mathrm{P(t)}\) equals approximately 32.9
2. INFER what t = 8 represents in context
- Since \(\mathrm{t}\) represents "years after 1984"
- When \(\mathrm{t = 8}\), we're looking at 8 years after 1984
- The year would be: \(\mathrm{1984 + 8 = 1992}\)
3. TRANSLATE the function value into real-world terms
- \(\mathrm{P(8) ≈ 32.9}\) means the population is approximately 32.9 million
- Since \(\mathrm{P(t)}\) gives population in millions
- Therefore: In 1992 (8 years after 1984), the predicted population was approximately 32.9 million
Answer: C. 8 years after 1984, the predicted population of this country was approximately 32.91 million.
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse which number represents the input variable and which represents the output.
They might think "\(\mathrm{P(8) = 32.9}\)" means either:
- 8 million people in some year, OR
- 32.9 years after some starting point
This leads them to select Choice A (8 million people) or Choice D (32.91 years after 1984) because they've reversed the roles of input and output in the function.
Second Most Common Error:
Poor contextual understanding: Students correctly identify that 8 is the input but forget that \(\mathrm{t}\) represents "years after 1984" rather than the actual year.
They might interpret \(\mathrm{P(8)}\) as referring to the year 1988 (thinking \(\mathrm{t}\) is the actual year) or get confused about what the base year represents. This leads to confusion about when exactly the 32.9 million population occurs, causing them to guess between the remaining choices.
The Bottom Line:
This problem requires careful attention to what each part of the function notation represents in the real-world context - the input variable, the output variable, and the reference point for time measurement.
In \(1984\), the predicted population of this country was approximately \(8\) million.
In \(1984\), the predicted population of this country was approximately \(32.91\) million.
\(8\) years after \(1984\), the predicted population of this country was approximately \(32.91\) million.
\(32.91\) years after \(1984\), the predicted population of this country was approximately \(8\) million.