A tech company models the number of active users of a new app by the function \(\mathrm{U(m) = 800 \cdot...
GMAT Advanced Math : (Adv_Math) Questions
A tech company models the number of active users of a new app by the function \(\mathrm{U(m) = 800 \cdot 2^{m/12}}\), where \(\mathrm{m}\) is the number of months since the app launched. Which of the following is the best interpretation of the number 800 in this context?
- The estimated number of active users at the time of launch
- The number of months between each doubling of active users
- The percent increase in active users each month
- The factor by which the number of active users multiplies every 12 months
- The increase in active users each month
The estimated number of active users at the time of launch
The number of months between each doubling of active users
The percent increase in active users each month
The factor by which the number of active users multiplies every 12 months
The increase in active users each month
1. TRANSLATE the function structure
- Given: \(\mathrm{U(m) = 800 \cdot 2^{(m/12)}}\) where \(\mathrm{m}\) = months since launch
- This follows the exponential form \(\mathrm{A \cdot b^{(t/k)}}\) where:
- \(\mathrm{A}\) = initial value coefficient
- \(\mathrm{b}\) = growth factor base
- \(\mathrm{t/k}\) = exponent with time scaling
2. INFER the strategy to find what 800 represents
- To identify what the coefficient 800 means, evaluate the function at the starting point
- At launch time: \(\mathrm{m = 0}\)
- This will isolate the coefficient from the exponential part
3. SIMPLIFY the evaluation at m = 0
- \(\mathrm{U(0) = 800 \cdot 2^{(0/12)}}\)
- \(\mathrm{U(0) = 800 \cdot 2^{0}}\)
- \(\mathrm{U(0) = 800 \cdot 1 = 800}\)
4. INFER the real-world meaning
- Since \(\mathrm{U(0) = 800}\), this represents the number of active users when \(\mathrm{m = 0}\)
- \(\mathrm{m = 0}\) corresponds to the launch time
- Therefore, 800 is the estimated number of active users at launch
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that they need to evaluate the function at a specific point to understand parameter meaning. Instead, they try to guess what 800 represents by looking at the other numbers in the function or making assumptions about exponential growth without systematic analysis.
This leads to confusion and guessing among the answer choices.
Second Most Common Error:
Conceptual confusion about exponential parameters: Students mix up which number represents which aspect of exponential growth. They might think 800 is the growth factor (confusing it with the base 2) or think it represents a time period (confusing it with the 12 in the exponent).
This may lead them to select Choice D (The factor by which the number of active users multiplies every 12 months) by incorrectly associating the largest number with the multiplication factor.
The Bottom Line:
This problem tests whether students understand that in exponential functions, you can isolate and interpret each parameter by strategic evaluation, particularly at t = 0 for the initial value.
The estimated number of active users at the time of launch
The number of months between each doubling of active users
The percent increase in active users each month
The factor by which the number of active users multiplies every 12 months
The increase in active users each month