The function \(\mathrm{g(t) = 85{,}000(0.75)^t}\) gives the number of bacteria in a culture t hours after antibiotic treatment begins, where...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{g(t) = 85{,}000(0.75)^t}\) gives the number of bacteria in a culture t hours after antibiotic treatment begins, where \(\mathrm{0 \leq t \leq 12}\). What is the best interpretation of the statement '\(\mathrm{g(3)}\) is approximately equal to \(\mathrm{35{,}859}\)' in this context?
\(\mathrm{3}\) hours after antibiotic treatment begins, the predicted bacteria count is approximately \(\mathrm{35,859}\).
\(\mathrm{3}\) hours after antibiotic treatment begins, the bacteria count will have decreased by a total of approximately \(\mathrm{35,859}\).
When the bacteria count is approximately \(\mathrm{35,859}\), it is \(\mathrm{3}\) times the bacteria count from the previous hour.
When the bacteria count is approximately \(\mathrm{35,859}\), it is \(\mathrm{75\%}\) of the bacteria count from the previous hour.
1. TRANSLATE the function and given information
- Given information:
- \(\mathrm{g(t) = 85,000(0.75)^t}\) represents bacteria count at time t hours
- We're told \(\mathrm{g(3) ≈ 35,859}\)
- What this tells us: We need to interpret what \(\mathrm{g(3)}\) means in this context
2. INFER what g(3) represents
- Since \(\mathrm{g(t)}\) gives bacteria count at time t hours after treatment begins
- \(\mathrm{g(3)}\) specifically gives bacteria count at time \(\mathrm{t = 3}\) hours
- Therefore: \(\mathrm{g(3) ≈ 35,859}\) means at 3 hours after treatment, there are about 35,859 bacteria
3. TRANSLATE this understanding to match answer choices
- Look for the choice that says: "at 3 hours after treatment, bacteria count is 35,859"
- This matches choice (A) exactly
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students focus on the number 35,859 and think it represents the decrease in bacteria rather than the actual count remaining.
They might reason: "Since bacteria are dying from antibiotics, 35,859 must be how many died." This leads them to select Choice B (decrease of 35,859).
Second Most Common Error:
Inadequate INFER reasoning: Students see the 0.75 decay factor and the number 3, then incorrectly connect these as "3 times" or focus on the "75%" relationship without understanding what \(\mathrm{g(3)}\) actually represents.
This confusion about relationships versus function evaluation may lead them to select Choice C (3 times) or Choice D (75% relationship).
The Bottom Line:
This problem tests whether students can distinguish between what a function outputs (the bacteria count) versus what causes that output (the decay process). The key insight is that \(\mathrm{g(3)}\) simply tells us the bacteria count at hour 3, not anything about the change or rate relationships.
\(\mathrm{3}\) hours after antibiotic treatment begins, the predicted bacteria count is approximately \(\mathrm{35,859}\).
\(\mathrm{3}\) hours after antibiotic treatment begins, the bacteria count will have decreased by a total of approximately \(\mathrm{35,859}\).
When the bacteria count is approximately \(\mathrm{35,859}\), it is \(\mathrm{3}\) times the bacteria count from the previous hour.
When the bacteria count is approximately \(\mathrm{35,859}\), it is \(\mathrm{75\%}\) of the bacteria count from the previous hour.