\(\mathrm{P(h) = 1000 \times 2^{h/3}}\) The function P gives the predicted population \(\mathrm{P(h)}\), in number of bacteria, of a certain...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{P(h) = 1000 \times 2^{h/3}}\)
The function P gives the predicted population \(\mathrm{P(h)}\), in number of bacteria, of a certain bacterial culture h hours after the initial measurement was taken in a laboratory, where \(\mathrm{h \leq 72}\). Which of the following is the best interpretation of the statement '\(\mathrm{P(24) \approx 256{,}000}\)' in this context?
The predicted population of the bacterial culture was approximately \(\mathrm{24}\) bacteria \(\mathrm{256{,}000}\) hours after the initial measurement.
The predicted population of the bacterial culture was approximately \(\mathrm{256{,}000}\) bacteria \(\mathrm{24}\) hours after the initial measurement.
The predicted population of the bacterial culture was approximately \(\mathrm{256{,}000}\) bacteria \(\mathrm{\frac{24}{3}}\) hours after the initial measurement.
The predicted population of the bacterial culture was approximately \(\mathrm{\frac{24}{3}}\) bacteria \(\mathrm{256{,}000}\) hours after the initial measurement.
1. TRANSLATE the function notation
- Given: \(\mathrm{P(h) = 1000 \times 2^{(h/3)}}\) where \(\mathrm{h \leq 72}\)
- This means:
- h is the input (hours after initial measurement)
- P(h) is the output (predicted bacterial population)
2. TRANSLATE the statement P(24) = 256,000
- \(\mathrm{P(24) = 256{,}000}\) means:
- When \(\mathrm{h = 24}\) (input 24 hours)
- \(\mathrm{P(h) = 256{,}000}\) (output is 256,000 bacteria)
3. INFER what this means in context
- The statement tells us the population prediction at a specific time
- 24 hours after the initial measurement → 256,000 bacteria predicted
- This matches the pattern: \(\mathrm{P(time) = population}\)
4. TRANSLATE back to plain English
- "The predicted population was 256,000 bacteria, 24 hours after the initial measurement"
- This matches choice (B) exactly
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students mix up which number represents what quantity in function notation.
They might think \(\mathrm{P(24) = 256{,}000}\) means "P of 256,000 equals 24" instead of "P of 24 equals 256,000." This leads them to interpret it as "24 bacteria at some time" rather than "256,000 bacteria at 24 hours."
This may lead them to select Choice A (24 bacteria at 256,000 hours).
Second Most Common Error:
Poor INFER reasoning: Students don't properly connect the function's mathematical structure to the real-world context.
They might recognize that \(\mathrm{24/3 = 8}\) appears in the exponent and think this 8-hour timepoint is somehow relevant to the interpretation, leading them to confuse when the population count occurs.
This may lead them to select Choice C (256,000 bacteria at 8 hours).
The Bottom Line:
Function notation interpretation requires carefully tracking which number is the input (goes inside parentheses) and which is the output (the result). The context tells you what each represents - here, hours go in, bacteria count comes out.
The predicted population of the bacterial culture was approximately \(\mathrm{24}\) bacteria \(\mathrm{256{,}000}\) hours after the initial measurement.
The predicted population of the bacterial culture was approximately \(\mathrm{256{,}000}\) bacteria \(\mathrm{24}\) hours after the initial measurement.
The predicted population of the bacterial culture was approximately \(\mathrm{256{,}000}\) bacteria \(\mathrm{\frac{24}{3}}\) hours after the initial measurement.
The predicted population of the bacterial culture was approximately \(\mathrm{\frac{24}{3}}\) bacteria \(\mathrm{256{,}000}\) hours after the initial measurement.