Question:C = 150e^(-0.12t)The equation gives the estimated concentration C, in milligrams per liter, of a medication in a patient's bloodstream...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{C = 150e^{-0.12t}}\)
The equation gives the estimated concentration C, in milligrams per liter, of a medication in a patient's bloodstream t hours after injection, where \(\mathrm{0 \leq t \leq 24}\). Which statement is the best interpretation of \(\mathrm{(t,C) = (3,105)}\) in this context?
- The medication's estimated concentration decreased 105 milligrams per liter every hour after injection.
- The medication's estimated concentration decreased 3 milligrams per liter every 105 hours after injection.
- 3 hours after injection, the medication's estimated concentration is 105 milligrams per liter.
- 105 hours after injection, the medication's estimated concentration is 3 milligrams per liter.
1. TRANSLATE the coordinate pair into context
- Given information:
- Function: \(\mathrm{C = 150e^{-0.12t}}\)
- \(\mathrm{t}\) = hours after injection
- \(\mathrm{C}\) = concentration in mg/L
- Coordinate pair: \(\mathrm{(t,C) = (3,105)}\)
- What this tells us:
- First coordinate (3) represents the \(\mathrm{t}\)-value: 3 hours after injection
- Second coordinate (105) represents the \(\mathrm{C}\)-value: 105 mg/L concentration
2. INFER what this point means
- This coordinate pair represents one specific moment in time
- It's NOT describing a rate of change or trend
- It's simply stating: "At this specific time (3 hours), the concentration is this specific value (105 mg/L)"
3. Match to answer choices
Looking at each option:
- (A) Talks about rate (105 mg/L per hour) - but we have a point, not a rate
- (B) Reverses the variables incorrectly (3 mg/L every 105 hours)
- (C) Correctly states: 3 hours after injection, concentration is 105 mg/L
- (D) Reverses the variables (105 hours, 3 mg/L)
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse which coordinate corresponds to which variable, switching \(\mathrm{t}\) and \(\mathrm{C}\) values.
They might think \(\mathrm{(3,105)}\) means \(\mathrm{t = 105}\) and \(\mathrm{C = 3}\), leading them to interpret this as "105 hours after injection, concentration is 3 mg/L."
This may lead them to select Choice D (105 hours after injection, concentration is 3 mg/L)
Second Most Common Error:
Poor INFER reasoning: Students misinterpret the coordinate pair as representing a rate of change rather than a specific point.
They see the numbers 3 and 105 and think about how concentration changes over time, rather than understanding this represents one moment. This can lead to confusion about "decrease per hour" language in choices A and B.
This leads to confusion and guessing between choices A and B.
The Bottom Line:
Coordinate pairs in context problems require careful attention to which variable corresponds to which coordinate position, and understanding that single points represent specific moments, not rates or trends.