\(\mathrm{N(d) = 115(0.90)^d}\). The function N defined above can be used to model the number of species of brachiopods at...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{N(d) = 115(0.90)^d}\). The function N defined above can be used to model the number of species of brachiopods at various ocean depths \(\mathrm{d}\), where \(\mathrm{d}\) is in hundreds of meters. Which of the following does the model predict?
For every increase in depth by 1 meter, the number of brachiopod species decreases by 115.
For every increase in depth by 1 meter, the number of brachiopod species decreases by 10%.
For every increase in depth by 100 meters, the number of brachiopod species decreases by 115.
For every increase in depth by 100 meters, the number of brachiopod species decreases by 10%.
1. TRANSLATE the function components
- Given: \(\mathrm{N(d) = 115(0.90)^d}\)
- 115 = initial number of species (at depth 0)
- 0.90 = base of exponential (less than 1, so decay)
- d = depth measured in hundreds of meters
2. INFER what type of change this represents
- Since the base (0.90) is between 0 and 1, this is exponential decay
- Each time d increases by 1, we multiply by 0.90
- This creates percentage change, not absolute change
3. SIMPLIFY the relationship between consecutive depths
- At depth d: \(\mathrm{N(d) = 115(0.90)^d}\)
- At depth d+1: \(\mathrm{N(d+1) = 115(0.90)^{(d+1)}}\)
- \(\mathrm{N(d+1) = 115(0.90)^d \times 0.90}\)
- \(\mathrm{N(d+1) = N(d) \times 0.90}\)
4. INFER what this multiplication means
- \(\mathrm{N(d+1) = 0.90 \times N(d)}\) means the new value is 90% of the old value
- If you have 90% remaining, you lost 10%
- So there's a 10% decrease
5. TRANSLATE the unit meaning
- Since d is in hundreds of meters, d+1 means depth increased by 100 meters
- Therefore: every 100-meter increase causes a 10% decrease
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't distinguish between exponential and linear change patterns.
They see "decreases" in the answer choices and think about absolute decreases rather than percentage decreases. This leads them to interpret the function as if it were linear (like \(\mathrm{N(d) = 115 - 10d}\)) instead of exponential. Since 115 appears in the function, they think each step decreases by 115.
This may lead them to select Choice A (decreases by 115 per meter) or Choice C (decreases by 115 per 100 meters).
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret the unit "hundreds of meters."
They correctly understand the 10% decrease but think it applies to each 1-meter increase instead of each 100-meter increase. They miss that d represents hundreds of meters, not individual meters.
This may lead them to select Choice B (10% decrease per meter).
The Bottom Line:
This problem tests whether students understand exponential functions create percentage changes (not absolute changes) and whether they can correctly interpret units in function variables. The key insight is recognizing that \(\mathrm{0.90^{(d+1)} = 0.90^d \times 0.90}\), showing each step retains 90% of the previous value.
For every increase in depth by 1 meter, the number of brachiopod species decreases by 115.
For every increase in depth by 1 meter, the number of brachiopod species decreases by 10%.
For every increase in depth by 100 meters, the number of brachiopod species decreases by 115.
For every increase in depth by 100 meters, the number of brachiopod species decreases by 10%.