A research group models the mass of a heated chemical sample with the function \(\mathrm{M(t) = 320(0.78)^t}\), where \(\mathrm{M(t)}\) is...
GMAT Advanced Math : (Adv_Math) Questions
A research group models the mass of a heated chemical sample with the function \(\mathrm{M(t) = 320(0.78)^t}\), where \(\mathrm{M(t)}\) is the predicted mass, in grams, remaining \(\mathrm{t}\) hours after heating begins and \(\mathrm{0 \lt t ≤ 8}\). If \(\mathrm{M(3) ≈ 151}\), which of the following is the best interpretation of this statement in this context?
- Three hours after heating begins, the predicted mass remaining is approximately 151 grams.
- Three hours after heating begins, the predicted mass remaining has decreased by a total of approximately 151 grams.
- When the predicted mass remaining is approximately 151 grams, it is 3 times the predicted mass 1 hour earlier.
- When the predicted mass remaining is approximately 151 grams, it is 22% less than the predicted mass 1 hour earlier.
Three hours after heating begins, the predicted mass remaining is approximately \(151\) grams.
Three hours after heating begins, the predicted mass remaining has decreased by a total of approximately \(151\) grams.
When the predicted mass remaining is approximately \(151\) grams, it is \(3\) times the predicted mass \(1\) hour earlier.
When the predicted mass remaining is approximately \(151\) grams, it is \(22\%\) less than the predicted mass \(1\) hour earlier.
1. TRANSLATE the given information
- Given: \(\mathrm{M(t) = 320(0.78)^t}\) represents remaining mass after t hours
- Given: \(\mathrm{M(3) \approx 151}\)
- This tells us: When \(\mathrm{t = 3}\), the function output is approximately 151
2. INFER what this means in context
- The input \(\mathrm{t = 3}\) represents "3 hours after heating begins"
- The output \(\mathrm{M(3) = 151}\) represents "the remaining mass is 151 grams"
- Therefore: At 3 hours after heating begins, approximately 151 grams remain
3. TRANSLATE this understanding to the answer choices
- Look for the choice that correctly states: "At 3 hours, 151 grams remain"
- This matches choice (A) exactly
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse what the function represents - remaining mass versus lost mass.
They see \(\mathrm{M(3) = 151}\) and think "151 grams have been lost by hour 3" rather than "151 grams remain at hour 3." This fundamental misreading of what the function represents leads them to select Choice B (151 grams decreased).
Second Most Common Error:
Poor INFER reasoning: Students misunderstand the role of the input value in function interpretation.
They see the "3" in \(\mathrm{M(3) = 151}\) and incorrectly think it means "3 times some other value" rather than recognizing it as the time input. This confusion about function notation leads them to select Choice C (3 times the previous mass).
The Bottom Line:
This problem tests whether students can correctly interpret function notation in context. The key insight is distinguishing between the input (time) and output (remaining quantity) of the function, and understanding what "remaining" means versus "lost."
Three hours after heating begins, the predicted mass remaining is approximately \(151\) grams.
Three hours after heating begins, the predicted mass remaining has decreased by a total of approximately \(151\) grams.
When the predicted mass remaining is approximately \(151\) grams, it is \(3\) times the predicted mass \(1\) hour earlier.
When the predicted mass remaining is approximately \(151\) grams, it is \(22\%\) less than the predicted mass \(1\) hour earlier.