Question:\(\mathrm{P(t) = 900(2.25)^{(t/2)}}\)The function P models the population of a certain species of algae in a pond, where \(\mathrm{P(t)}\) is...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{P(t) = 900(2.25)^{(t/2)}}\)
The function P models the population of a certain species of algae in a pond, where \(\mathrm{P(t)}\) is the population after a period of \(\mathrm{t}\) years. The population increases by \(\mathrm{k\%}\) every 2 years. What is the value of k?
1. TRANSLATE the problem information
- Given: \(\mathrm{P(t) = 900(2.25)^{(t/2)}}\) where t is in years
- Find: k, the percentage increase every 2 years
2. INFER the meaning of the exponent structure
- The exponent is \(\mathrm{t/2}\), not just t
- This means when t increases by 2 years, the exponent increases by 1
- So every 2 years, we multiply by \(2.25\) exactly once
- Therefore: \(2.25\) is the growth factor for each 2-year period
3. SIMPLIFY to find the growth factor relationship
- Growth Factor = 1 + rate (as decimal)
- We have: \(2.25 = 1 + r\)
- Solving: \(r = 2.25 - 1 = 1.25\)
4. TRANSLATE decimal rate to percentage
- Rate as decimal: \(r = 1.25\)
- Rate as percentage: \(\mathrm{k = 1.25 \times 100 = 125\%}\)
Answer: C) 125
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students see 2.25 in the function and assume it directly represents the percentage increase.
They think: "The base is 2.25, so the percentage must be 225%." This skips the crucial step of understanding that growth factor = 1 + rate, not just the rate itself. The base 2.25 means the population becomes 2.25 times what it was, which represents a 125% increase (since \(2.25 = 1 + 1.25\)).
This may lead them to select Choice D (225).
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly find that \(r = 2.25 - 1 = 1.25\), but then think this decimal directly represents the percentage.
They conclude: "The rate is 1.25, so k = 1.25%." This misses the conversion step from decimal form (1.25) to percentage form (1.25 × 100 = 125%).
This leads to confusion since 1.25 isn't among the answer choices, causing them to guess or select the closest value.
The Bottom Line:
This problem requires understanding two key relationships: how exponent structure relates to time periods, and how growth factors connect to percentage rates. Students who rush past either connection will struggle to reach the correct answer systematically.