Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A park ranger is tracking the population of a certain species of bird. The population P(t) after t years is modeled by the function P(t) = 5000 / (1 + 4e^(-0.5t)). What is the limiting value (carrying capacity) of the bird population according to this model?
- A20000 birds
- B500 birds
- C5000 birds
- D1000 birds
Show answer & explanationAnswer & explanation
Correct answer: C. 5000 birds
This is a logistic growth model. As t approaches infinity, the term e^(-0.5t) approaches 0. Therefore, the denominator approaches 1 + 4(0) = 1. So, P(t) approaches 5000 / 1 = 5000. This upper limit is the carrying capacity.
Why the other options are wrong
- A. Incorrect. This would be 5000 * 4, an error in understanding the role of the denominator.
- B. Incorrect. This would be 5000 / 10 if there was an error in the denominator approaching 0.
- D. Incorrect. This would be 5000 / 5 if the 4e^(-0.5t) term approached 4.
Logistic Growth Model Limiting Value
In a logistic growth function of the form P(t) = K / (1 + Ae^(-rt)), the limiting value or carrying capacity (K) is the maximum population that the environment can sustain, represented by the numerator.
- Models population growth with a cap.
- Limiting value is the numerator (K).
- Occurs as t approaches infinity.
Memory trick: The 'K' in K/(1+Ae^-rt) is like the 'K'apacity of the environment.