A laptop's resale value, in dollars, is modeled by \(\mathrm{V(t) = 1,500(0.7)^t}\), where t is the number of years since...
GMAT Advanced Math : (Adv_Math) Questions
A laptop's resale value, in dollars, is modeled by \(\mathrm{V(t) = 1,500(0.7)^t}\), where \(\mathrm{t}\) is the number of years since the laptop was purchased and \(\mathrm{0 ≤ t ≤ 4}\). If \(\mathrm{y = V(t)}\) 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 minimum estimated resale value during the 4-year period was $1,500.
The estimated resale value at the time of purchase was about $360.
The model predicts the resale value decreases by $1,500 each year.
The estimated resale value at the time of purchase was $1,500.
1. TRANSLATE the y-intercept concept to mathematics
- Given information:
- Function: \(\mathrm{V(t) = 1,500(0.7)^t}\)
- Need to interpret the y-intercept of y = V(t)
- What this tells us: The y-intercept occurs when the input variable equals zero, so we need \(\mathrm{V(0)}\)
2. SIMPLIFY to find the y-intercept value
- Substitute t = 0 into the function:
\(\mathrm{V(0) = 1,500(0.7)^0}\) - Since any number raised to the power of 0 equals 1:
\(\mathrm{V(0) = 1,500(1) = 1,500}\)
3. INFER the contextual meaning
- Since t represents "years since the laptop was purchased"
- When t = 0, this means 0 years since purchase = at the time of purchase
- Therefore, \(\mathrm{V(0) = \$1,500}\) represents the resale value at the time of purchase
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may not connect "y-intercept" with "evaluate the function at t = 0"
Some students think the y-intercept is just the coefficient 1,500 without understanding they need to evaluate V(0). Others might confuse the y-intercept with other important values like the minimum or maximum, leading them to select Choice A ($1,500) for the wrong reason or get confused about which value represents what.
Second Most Common Error:
Poor INFER reasoning about context: Students correctly find V(0) = 1,500 but misunderstand what t = 0 means
They might think t = 0 represents some other time point or confuse it with the end of the time period. This contextual confusion can cause them to select Choice B (about $360) if they calculate \(\mathrm{V(4)}\) instead, thinking that's what the question is asking for.
The Bottom Line:
This problem tests whether students can connect the mathematical concept of y-intercept with function evaluation and then interpret that mathematical result within the given real-world context.
The minimum estimated resale value during the 4-year period was $1,500.
The estimated resale value at the time of purchase was about $360.
The model predicts the resale value decreases by $1,500 each year.
The estimated resale value at the time of purchase was $1,500.