A medication amount in a patient's bloodstream is modeled by \(\mathrm{M(t) = 3500(0.85)^t}\), where \(\mathrm{M(t)}\) is measured in milligrams and...
GMAT Advanced Math : (Adv_Math) Questions
A medication amount in a patient's bloodstream is modeled by \(\mathrm{M(t) = 3500(0.85)^t}\), where \(\mathrm{M(t)}\) is measured in milligrams and t is the number of hours after a single dose. Which of the following best interprets the number 3500 in this model?
- The initial amount, in milligrams, of the medication in the bloodstream
- The amount, in milligrams, eliminated each hour
- The amount, in milligrams, remaining after 1 hour
- The percent of the medication that remains each hour
The initial amount, in milligrams, of the medication in the bloodstream
The amount, in milligrams, eliminated each hour
The amount, in milligrams, remaining after 1 hour
The percent of the medication that remains each hour
1. TRANSLATE the problem information
- Given: \(\mathrm{M(t) = 3500(0.85)^t}\) models medication amount
- \(\mathrm{M(t)}\) = medication in milligrams, \(\mathrm{t}\) = hours after dose
- Question: What does 3500 represent?
2. INFER the approach
- In exponential models \(\mathrm{A \cdot b^t}\), the coefficient A represents the initial value
- To verify this, evaluate the function when \(\mathrm{t = 0}\) (initial moment)
3. SIMPLIFY by evaluating M(0)
- \(\mathrm{M(0) = 3500(0.85)^0}\)
- Since any number to the 0 power equals 1: \(\mathrm{(0.85)^0 = 1}\)
- Therefore: \(\mathrm{M(0) = 3500(1) = 3500}\) milligrams
4. INFER the meaning
- At \(\mathrm{t = 0}\) (immediately after dose), there are 3500 mg in the bloodstream
- This confirms 3500 is the initial amount of medication
Answer: A. The initial amount, in milligrams, of the medication in the bloodstream
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 \(\mathrm{t = 0}\) to interpret the coefficient. Instead, they might guess based on the units or try to relate 3500 to other given values without systematic analysis. This leads to confusion and guessing among all answer choices.
Second Most Common Error:
Conceptual confusion about exponential models: Students might confuse what 0.85 represents versus what 3500 represents, thinking 3500 might be related to the percentage or rate of change rather than the initial value. This may lead them to select Choice D (The percent of the medication that remains each hour).
The Bottom Line:
Students often struggle to connect the abstract mathematical form \(\mathrm{A \cdot b^t}\) with its real-world meaning, especially identifying which parameter represents the initial condition versus the growth/decay factor.
The initial amount, in milligrams, of the medication in the bloodstream
The amount, in milligrams, eliminated each hour
The amount, in milligrams, remaining after 1 hour
The percent of the medication that remains each hour