The function \(\mathrm{C(t) = 80(0.82)^t}\) models the concentration, in micrograms per milliliter (μg/mL), of a certain drug in a patient's...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{C(t) = 80(0.82)^t}\) models the concentration, in micrograms per milliliter (\(\mathrm{μg/mL}\)), of a certain drug in a patient's bloodstream \(\mathrm{t}\) hours after it was administered. What is the best interpretation of the statement '\(\mathrm{C(3) \approx 44.1}\)' in this context?
1. TRANSLATE the function components
- Given information:
- Function: \(\mathrm{C(t) = 80(0.82)^t}\)
- \(\mathrm{C(t)}\) = drug concentration in \(\mathrm{μg/mL}\)
- \(\mathrm{t}\) = hours after drug administration
- Statement: \(\mathrm{C(3) ≈ 44.1}\)
2. INFER the function notation structure
- In function notation \(\mathrm{C(t)}\):
- The value inside parentheses (3) is the input
- The result of the function (44.1) is the output
- So \(\mathrm{C(3)}\) asks: "What's the concentration at \(\mathrm{t = 3}\) hours?"
3. TRANSLATE the mathematical statement to English
- \(\mathrm{C(3) ≈ 44.1}\) breaks down as:
- Input: \(\mathrm{t = 3}\) (hours after administration)
- Output: 44.1 (concentration in \(\mathrm{μg/mL}\))
- Translation: "3 hours after the drug was administered, its concentration is approximately 44.1 \(\mathrm{μg/mL}\)"
4. Match to answer choices
- Choice A matches our translation exactly
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse which number represents time versus concentration
They might think "44.1 hours" or "3 \(\mathrm{μg/mL}\)" because they don't carefully track units or mix up input/output positions. This confusion leads them to select Choice D (It takes approximately 44.1 hours for the drug's concentration to reach a level of 3 \(\mathrm{μg/mL}\)) where the numbers are completely reversed.
Second Most Common Error:
Poor INFER reasoning about function meaning: Students think \(\mathrm{C(3) ≈ 44.1}\) describes a change or decrease rather than an absolute value
They might interpret this as "the concentration decreased by 44.1" instead of "the concentration equals 44.1." This leads them to select Choice B which incorrectly focuses on decrease rather than absolute concentration level.
The Bottom Line:
This problem tests whether students truly understand function notation as an input-output system, not just symbol manipulation. The key insight is that mathematical statements must be translated with careful attention to units and variable meanings.