Question:After a promotional campaign for a mobile app ends, the function \(\mathrm{U(t) = 1{,}800(0.92)^t}\) models the number of active users...
GMAT Advanced Math : (Adv_Math) Questions
After a promotional campaign for a mobile app ends, the function \(\mathrm{U(t) = 1{,}800(0.92)^t}\) models the number of active users of the app at the end of each week. In this model, \(\mathrm{t}\) represents the number of weeks since the campaign ended, and \(\mathrm{U(t)}\) gives the estimated number of active users at that time. Which of the following is the best interpretation of \(\mathrm{U(3) \approx 1{,}400}\) in this context?
The number of active users is estimated to be about 1,400 at the end of the third week after the campaign ended.
The number of active users is estimated to decrease by about 1,400 each week during the first three weeks after the campaign.
The number of active users is estimated to be about three times the initial number at the end of the third week.
The number of active users is estimated to be about 1,400 greater in week 3 than in week 0.
1. TRANSLATE the function information
- Given information:
- \(\mathrm{U(t) = 1,800(0.92)^t}\) models active users
- \(\mathrm{t}\) = weeks since campaign ended
- \(\mathrm{U(t)}\) = number of active users at time t
- We know \(\mathrm{U(3) \approx 1,400}\)
- What this tells us: \(\mathrm{U(3)}\) means "the number of active users when \(\mathrm{t = 3}\)"
2. INFER what U(3) represents in context
- Since t represents weeks since the campaign ended
- \(\mathrm{U(3)}\) = the number of active users 3 weeks after the campaign ended
- The value 1,400 is the actual count of users at that specific time
3. TRANSLATE each answer choice to check interpretation
- (A): \(\mathrm{U(3)}\) = number of users at end of week 3 ✓
- (B): \(\mathrm{U(3)}\) = weekly decrease amount ✗ (This would be a rate, not a count)
- (C): \(\mathrm{U(3)}\) = 3 times initial number ✗ (This would be \(\mathrm{3 \times U(0)}\), not \(\mathrm{U(3)}\))
- (D): \(\mathrm{U(3)}\) = difference from week 0 ✗ (This would be \(\mathrm{U(3) - U(0)}\), not \(\mathrm{U(3)}\))
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse what the function output represents versus what the input represents.
They might think \(\mathrm{U(3) \approx 1,400}\) means something about the number 3 (like "3 times something" or "decrease over 3 weeks") rather than understanding it means "the output when the input is 3."
This may lead them to select Choice C (thinking about multiplication by 3) or Choice B (thinking about change over 3 weeks).
Second Most Common Error:
Inadequate INFER reasoning: Students understand \(\mathrm{U(3)}\) gives a value at week 3, but misinterpret whether it's an absolute amount versus a change from the starting point.
They know it's about week 3 but think \(\mathrm{U(3)}\) represents how much the users changed compared to the beginning, rather than the total number of users at that time.
This may lead them to select Choice D (interpreting as an increase from week 0).
The Bottom Line:
Function notation problems require careful attention to what the input and output represent. The output \(\mathrm{U(3)}\) is always the actual function value at that input, not a rate of change, not a multiple of the input, and not a difference from another point.
The number of active users is estimated to be about 1,400 at the end of the third week after the campaign ended.
The number of active users is estimated to decrease by about 1,400 each week during the first three weeks after the campaign.
The number of active users is estimated to be about three times the initial number at the end of the third week.
The number of active users is estimated to be about 1,400 greater in week 3 than in week 0.