A pharmacist models the concentration of a medication in a patient's bloodstream using the function \(\mathrm{g(t) = 120(0.8)^{t}}\), where \(\mathrm{...
GMAT Advanced Math : (Adv_Math) Questions
A pharmacist models the concentration of a medication in a patient's bloodstream using the function \(\mathrm{g(t) = 120(0.8)^{t}}\), where \(\mathrm{g(t)}\) is measured in micrograms per milliliter. In this function, \(\mathrm{t}\) represents the number of hours after the medication is administered, where \(\mathrm{t \leq 8}\). Which of the following is the best interpretation of \(\mathrm{120}\) in this context?
The estimated initial concentration of the medication in the patient's bloodstream when first administered
The estimated concentration of the medication in the patient's bloodstream 8 hours after administration
The estimated percent decrease in medication concentration in the patient's bloodstream each hour
The estimated percent decrease in medication concentration in the patient's bloodstream over 8 hours
1. TRANSLATE the problem information
- Given: \(\mathrm{g(t) = 120(0.8)^t}\) represents medication concentration
- Find: What does 120 represent in this context?
- Context: t is hours after administration, g(t) is concentration in micrograms per milliliter
2. INFER the mathematical relationship
- This is an exponential function in the form \(\mathrm{f(x) = ab^x}\)
- In exponential functions, the coefficient 'a' always represents the initial value when the input equals zero
- So I need to find what happens when \(\mathrm{t = 0}\) (at the moment of administration)
3. SIMPLIFY the evaluation at t = 0
- Substitute \(\mathrm{t = 0}\): \(\mathrm{g(0) = 120(0.8)^0}\)
- Since any number to the power 0 equals 1: \(\mathrm{(0.8)^0 = 1}\)
- Therefore: \(\mathrm{g(0) = 120(1) = 120}\)
4. TRANSLATE the mathematical result back to context
- At \(\mathrm{t = 0}\) (when medication is first administered), the concentration is 120 micrograms per milliliter
- This means 120 represents the initial concentration
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Conceptual confusion about exponential function components: Students often confuse what the base (0.8) versus the coefficient (120) represents in exponential decay models.
They might think 120 has something to do with the decay rate since it's a "big number," leading them to incorrectly select Choice C or Choice D related to percentage decrease. They don't recognize that 120 is simply the starting amount.
Second Most Common Error:
Weak INFER skill: Students don't connect the mathematical structure \(\mathrm{f(x) = ab^x}\) with the real-world meaning of "initial value."
They might not think to substitute \(\mathrm{t = 0}\) to find what the coefficient represents, instead trying to work backwards from the answer choices or guessing based on which number "looks right" in context. This leads to confusion and random answer selection.
The Bottom Line:
This problem requires understanding the standard form of exponential functions and recognizing that coefficients represent initial values. Success depends on knowing to substitute the input variable equal to zero and interpreting the result in context.
The estimated initial concentration of the medication in the patient's bloodstream when first administered
The estimated concentration of the medication in the patient's bloodstream 8 hours after administration
The estimated percent decrease in medication concentration in the patient's bloodstream each hour
The estimated percent decrease in medication concentration in the patient's bloodstream over 8 hours