A conservation group models the number of trees in a new plantation with the function \(\mathrm{T(y) = 320(1.08)}^{(\mathrm{y}/2)}\), where y...
GMAT Advanced Math : (Adv_Math) Questions
A conservation group models the number of trees in a new plantation with the function \(\mathrm{T(y) = 320(1.08)}^{(\mathrm{y}/2)}\), where y is the number of years since planting. Which of the following statements is best supported by this model?
For each additional year, the number of trees increases by \(\mathrm{320}\).
For each additional year, the number of trees increases by \(\mathrm{8\%}\).
For each additional \(\mathrm{2}\) years, the number of trees increases by \(\mathrm{320}\).
For each additional \(\mathrm{2}\) years, the number of trees increases by \(\mathrm{8\%}\).
For each additional \(\mathrm{2}\) years, the number of trees decreases by \(\mathrm{8\%}\).
1. TRANSLATE the problem information
- Given: \(\mathrm{T(y) = 320(1.08)^{(y/2)}}\) models tree population after y years
- Need to determine: Which statement about the growth pattern is correct
- Key insight: The answer choices focus on yearly vs. 2-year intervals and different growth amounts
2. INFER the approach needed
- Since the function has y/2 in the exponent, changes in y by 2 units will be more straightforward to analyze than changes by 1 unit
- To test each statement, I need to see what happens to T(y) when y increases by the specified time periods
- The key is understanding how the exponent changes when the input changes
3. SIMPLIFY the analysis for 2-year intervals
- When y increases by 2: \(\mathrm{T(y+2) = 320(1.08)^{((y+2)/2)}}\)
- SIMPLIFY the exponent: \(\mathrm{(y+2)/2 = y/2 + 1}\)
- So \(\mathrm{T(y+2) = 320(1.08)^{(y/2 + 1)} = 320(1.08)^{(y/2)} \times (1.08)^1}\)
- Therefore: \(\mathrm{T(y+2) = T(y) \times 1.08}\)
This means every 2 years, the population is multiplied by 1.08, which is exactly an 8% increase.
4. SIMPLIFY the analysis for 1-year intervals (to eliminate wrong answers)
- When y increases by 1: \(\mathrm{T(y+1) = 320(1.08)^{((y+1)/2)}}\)
- SIMPLIFY the exponent: \(\mathrm{(y+1)/2 = y/2 + 1/2}\)
- So \(\mathrm{T(y+1) = T(y) \times (1.08)^{(1/2)} = T(y) \times (1.08)^{0.5}}\)
- Using calculator: \(\mathrm{(1.08)^{0.5} \approx 1.039}\), which is about 3.9% increase per year
5. INFER which answer choice matches
- Choices A and C suggest adding 320 each time period (linear growth), but this is exponential growth
- Choice B suggests 8% increase per year, but we found ~3.9% per year
- Choice E suggests decrease, but \(\mathrm{1.08 \gt 1}\) means growth
- Choice D matches our finding: 8% increase every 2 years
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students often misinterpret the coefficient 320 as the amount added per time period, treating this as linear rather than exponential growth.
They see \(\mathrm{T(y) = 320(1.08)^{(y/2)}}\) and think "320 trees are added each year" because 320 is prominently displayed. This linear thinking makes them gravitate toward choices A or C, selecting Choice A (320 increase per year) without recognizing that 320 is the initial population, not the growth amount.
Second Most Common Error:
Inadequate SIMPLIFY execution: Students correctly identify this as exponential growth but make algebraic errors when analyzing the exponent changes.
They might incorrectly calculate T(y+1) or T(y+2), leading to wrong growth percentages. Some get confused by the y/2 structure and mistakenly think each year corresponds to an 8% increase, selecting Choice B (8% per year) instead of recognizing that the 8% applies to 2-year intervals.
The Bottom Line:
This problem tests whether students can distinguish between the initial value and growth rate in exponential functions, and whether they can correctly analyze how fractional exponents behave under input changes.
For each additional year, the number of trees increases by \(\mathrm{320}\).
For each additional year, the number of trees increases by \(\mathrm{8\%}\).
For each additional \(\mathrm{2}\) years, the number of trees increases by \(\mathrm{320}\).
For each additional \(\mathrm{2}\) years, the number of trees increases by \(\mathrm{8\%}\).
For each additional \(\mathrm{2}\) years, the number of trees decreases by \(\mathrm{8\%}\).