A biologist models the population of a species in a protected area. The function \(\mathrm{P(t) = 50(2)^{(t/8)}}\) gives the predicted...
GMAT Advanced Math : (Adv_Math) Questions
A biologist models the population of a species in a protected area. The function \(\mathrm{P(t) = 50(2)^{(t/8)}}\) gives the predicted number of individuals of the species t years after the start of the observation. Which of the following is the best interpretation of the statement \(\mathrm{P(24) = 400}\) in this context?
Approximately 24 years after the start of the observation, the population is predicted to be 400.
The population is predicted to increase by 400 individuals in the 24 years after the observation starts.
The population is predicted to be 24 individuals approximately 400 years after the observation starts.
The initial population of the species was 24, and it is predicted to grow to a final population of 400.
1. TRANSLATE the function and variables
- Given information:
- \(\mathrm{P(t) = 50(2)^{(t/8)}}\) models population
- \(\mathrm{t}\) = years after observation starts
- \(\mathrm{P(t)}\) = predicted number of individuals
- We need to interpret \(\mathrm{P(24) = 400}\)
2. TRANSLATE the mathematical statement
- \(\mathrm{P(24) = 400}\) means:
- Input: \(\mathrm{t = 24}\)
- Output: \(\mathrm{P(t) = 400}\)
- In words: "When t is 24 years, P(t) is 400 individuals"
3. INFER the contextual meaning
- Since t represents "years after observation starts":
- \(\mathrm{t = 24}\) means "24 years after observation starts"
- Since P(t) represents "predicted population":
- \(\mathrm{P(t) = 400}\) means "population is predicted to be 400"
4. TRANSLATE back to full context
- Complete interpretation: "24 years after the start of observation, the population is predicted to be 400"
- This matches Choice A exactly
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse what the numbers in \(\mathrm{P(24) = 400}\) represent, often swapping the input and output values.
They might think "\(\mathrm{P(24) = 400}\)" means the population is 24 when the time is 400 years, leading them to select Choice C (The population is predicted to be 24 individuals approximately 400 years after the observation starts).
Second Most Common Error:
Conceptual confusion about function outputs: Students misinterpret 400 as representing a change in population rather than the total population value.
They calculate that since \(\mathrm{P(0) = 50}\), an "increase of 400" seems reasonable over 24 years, leading them to select Choice B (The population is predicted to increase by 400 individuals in the 24 years after the observation starts).
The Bottom Line:
This problem requires precise interpretation of function notation. Students must clearly distinguish between inputs (independent variable) and outputs (dependent variable), then accurately translate mathematical statements back into contextual language.
Approximately 24 years after the start of the observation, the population is predicted to be 400.
The population is predicted to increase by 400 individuals in the 24 years after the observation starts.
The population is predicted to be 24 individuals approximately 400 years after the observation starts.
The initial population of the species was 24, and it is predicted to grow to a final population of 400.