\(\mathrm{P(t) = 2{,}500(1.6)^t}\) The given function P models the estimated population of Northwood, where t represents the number of years...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{P(t) = 2{,}500(1.6)^t}\)
The given function P models the estimated population of Northwood, where t represents the number of years since the beginning of 2015, and \(\mathrm{0 ≤ t ≤ 3}\). If \(\mathrm{y = P(t)}\) is graphed in the ty-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?
- The maximum estimated population of Northwood during the 4-year period was 2,500.
- The maximum estimated population of Northwood during the 4-year period was 10,200.
- The estimated population of Northwood at the beginning of 2015 was 2,500.
- The estimated population of Northwood at the beginning of 2015 was 10,200.
The maximum estimated population of Northwood during the \(4\)-year period was \(2,500\).
The maximum estimated population of Northwood during the \(4\)-year period was \(10,200\).
The estimated population of Northwood at the beginning of \(2015\) was \(2,500\).
The estimated population of Northwood at the beginning of \(2015\) was \(10,200\).
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{P(t) = 2,500(1.6)^t}\) models population
- \(\mathrm{t}\) = years since beginning of 2015
- Need to interpret y-intercept of \(\mathrm{y = P(t)}\)
- What this tells us: The y-intercept occurs where the graph crosses the y-axis (when \(\mathrm{t = 0}\))
2. INFER what the y-intercept means
- Since \(\mathrm{t = 0}\) represents the beginning of 2015, the y-intercept tells us the population at that starting point
- To find this value, we need to evaluate \(\mathrm{P(0)}\)
3. SIMPLIFY to find the y-intercept value
- Substitute \(\mathrm{t = 0}\) into the function:
\(\mathrm{P(0) = 2,500(1.6)^0}\) - Apply zero exponent rule:
\(\mathrm{P(0) = 2,500(1) = 2,500}\)
4. TRANSLATE back to context
- The y-intercept value of 2,500 represents the estimated population of Northwood at the beginning of 2015
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse y-intercept with maximum value of the function.
Since this is an exponential growth function with base 1.6 > 1, students might think the y-intercept represents the maximum population during the 4-year period. They calculate \(\mathrm{P(0) = 2,500}\) correctly but then select Choice A (maximum was 2,500) instead of recognizing this as the starting population value.
Second Most Common Error:
Poor INFER reasoning: Students calculate the maximum value instead of the y-intercept.
They recognize that for exponential growth, the maximum occurs at \(\mathrm{t = 3}\), so they calculate \(\mathrm{P(3) = 2,500(1.6)^3}\)
\(\mathrm{= 2,500(4.096) = 10,240 ≈ 10,200}\). This leads them to select Choice B (maximum was 10,200) or Choice D (beginning population was 10,200), completely missing what the y-intercept actually represents.
The Bottom Line:
The key challenge is distinguishing between what the y-intercept represents (starting value when \(\mathrm{t = 0}\)) versus other characteristics of exponential functions (like maximum values). Success requires careful contextual interpretation, not just computational skills.
The maximum estimated population of Northwood during the \(4\)-year period was \(2,500\).
The maximum estimated population of Northwood during the \(4\)-year period was \(10,200\).
The estimated population of Northwood at the beginning of \(2015\) was \(2,500\).
The estimated population of Northwood at the beginning of \(2015\) was \(10,200\).