\(\mathrm{P(x) = 250(1.08)^x}\) The function P above models the population of a certain species of fish in a lake, where...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{P(x) = 250(1.08)^x}\)
The function P above models the population of a certain species of fish in a lake, where \(\mathrm{x}\) is the number of years after the end of 2010, and \(\mathrm{0 ≤ x ≤ 6}\). If \(\mathrm{y = P(x)}\) is graphed in the \(\mathrm{xy}\)-plane, which of the following is the best interpretation of the \(\mathrm{y}\)-intercept of the graph in this context?
The estimated population of the fish increases by \(8\%\) each year.
The minimum estimated population of the fish during the \(6\) years was \(250\).
The estimated population of the fish at the end of \(2010\) was \(250\).
The estimated population of the fish at the end of \(2011\) was \(270\).
1. TRANSLATE the y-intercept concept to mathematical terms
- The y-intercept occurs when \(\mathrm{x = 0}\)
- We need to find \(\mathrm{P(0)}\) for the function \(\mathrm{P(x) = 250(1.08)^x}\)
2. SIMPLIFY to find the y-intercept value
- \(\mathrm{P(0) = 250(1.08)^0}\)
- Since any non-zero number to the power of 0 equals 1: \(\mathrm{(1.08)^0 = 1}\)
- Therefore: \(\mathrm{P(0) = 250(1) = 250}\)
- The y-intercept is the point \(\mathrm{(0, 250)}\)
3. TRANSLATE the time reference to understand what x = 0 means
- The problem states: "x is the number of years after the end of 2010"
- So \(\mathrm{x = 0}\) corresponds to "the end of 2010"
4. INFER the contextual meaning
- The y-intercept \(\mathrm{(0, 250)}\) means that when \(\mathrm{x = 0}\) (at the end of 2010), the fish population \(\mathrm{P(x) = 250}\)
- This represents the initial fish population in our model
Answer: C - The estimated population of the fish at the end of 2010 was 250.
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students correctly find that \(\mathrm{P(0) = 250}\), but misinterpret the time reference. They might think \(\mathrm{x = 0}\) corresponds to "the beginning of 2010" or "2010 in general" rather than specifically "the end of 2010."
This confusion about the time reference can lead them to hesitate between answer choices or select an incorrect interpretation of when the population was 250.
Second Most Common Error:
Conceptual confusion about exponential functions: Students might focus on the growth rate (1.08 representing 8% increase) instead of identifying what the y-intercept represents. They see "8%" in the function and immediately jump to choice (A) without properly analyzing what the y-intercept means.
This may lead them to select Choice A (The estimated population increases by 8% each year).
The Bottom Line:
This problem requires careful attention to both the mathematical concept of y-intercept and precise interpretation of the time reference. Success depends on translating "y-intercept" to "\(\mathrm{x = 0}\)" and then correctly interpreting what "\(\mathrm{x = 0}\)" means in the given context.
The estimated population of the fish increases by \(8\%\) each year.
The minimum estimated population of the fish during the \(6\) years was \(250\).
The estimated population of the fish at the end of \(2010\) was \(250\).
The estimated population of the fish at the end of \(2011\) was \(270\).