A streaming service releases a major app update. The function \(\mathrm{N(t) = 5000(0.84)^t}\), where 0 leq t leq 8, gives...
GMAT Advanced Math : (Adv_Math) Questions
A streaming service releases a major app update. The function \(\mathrm{N(t) = 5000(0.84)^t}\), where \(\mathrm{0 \leq t \leq 8}\), gives the predicted number of active daily users t weeks after the update. A researcher states that \(\mathrm{N(4)}\) is approximately equal to \(\mathrm{2{,}500}\). Which of the following is the best interpretation of this statement in context?
When the predicted number of active users is approximately 2,500, it is decreasing by 4% per week.
When the predicted number of active users is approximately 2,500, it is 4 times the predicted number of active users in the previous week.
From the day of the update to 4 weeks after the update, the predicted number of active users decreased by a total of approximately 2,500 users.
Four weeks after the update, the predicted number of active users is approximately 2,500.
1. TRANSLATE the mathematical statement
- Given information:
- \(\mathrm{N(t) = 5000(0.84)^t}\) predicts active daily users t weeks after update
- Statement: \(\mathrm{N(4) ≈ 2,500}\)
- What this tells us: We need to interpret what \(\mathrm{N(4)}\) means in the context of this problem
2. INFER the meaning of function notation
- \(\mathrm{N(4)}\) means "evaluate the function N when the input t equals 4"
- Since t represents "weeks after the update," \(\mathrm{N(4)}\) represents the predicted number of users exactly 4 weeks after the update
- The value 2,500 is the output of the function, not a rate or change
3. TRANSLATE this into plain English
- \(\mathrm{N(4) ≈ 2,500}\) means: "Four weeks after the update, the predicted number of active daily users is approximately 2,500"
4. INFER which answer choice matches this interpretation
- Scan the choices for one that directly states this interpretation
- Choice (D) states exactly what we determined
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret the "4" in \(\mathrm{N(4)}\) as representing something other than the input value.
Many students see "4" and think it must relate to the 4% decay rate (since \(\mathrm{0.84 = 1 - 0.16}\), which is 16% decay, but students might confuse this). They incorrectly connect the "4" to a percentage, leading them to select Choice A (decreasing by 4% per week).
Second Most Common Error:
Poor function notation understanding: Students don't recognize that \(\mathrm{N(4)}\) represents a function evaluation and instead interpret "4" as a multiplication factor.
This conceptual confusion about function notation causes them to think \(\mathrm{N(4)}\) means "4 times" something, leading them to select Choice B (4 times the predicted number).
The Bottom Line:
Function notation in context problems requires careful translation between mathematical symbols and real-world meanings. The key insight is recognizing that \(\mathrm{N(4)}\) simply means "plug 4 into the function" - it's the predicted value when t = 4 weeks.
When the predicted number of active users is approximately 2,500, it is decreasing by 4% per week.
When the predicted number of active users is approximately 2,500, it is 4 times the predicted number of active users in the previous week.
From the day of the update to 4 weeks after the update, the predicted number of active users decreased by a total of approximately 2,500 users.
Four weeks after the update, the predicted number of active users is approximately 2,500.