Digital SATMath: AlgebraMedium
A scientist is tracking the population of a certain type of insect. The population, P(t), after t months, is modeled by the function P(t) = 500 / (1 + 4 * e^(-0.2t)). What is the carrying capacity of this insect population?
- A100 insects
- B200 insects
- C2000 insects
- D500 insects
Show answer & explanationAnswer & explanation
Correct answer: D. 500 insects
This is a logistic growth model, which has the general form P(t) = K / (1 + A * e^(-rt)). In this model, K represents the carrying capacity. By comparing the given function P(t) = 500 / (1 + 4 * e^(-0.2t)) to the general form, we can see that K = 500. The carrying capacity is the maximum population the environment can sustain, which is the asymptote of the function as t approaches infinity.
Why the other options are wrong
- A. Incorrect. This is not directly derivable from the logistic model parameters in this way.
- B. Incorrect. This is not directly derivable from the logistic model parameters in this way.
- C. Incorrect. This is not directly derivable from the logistic model parameters in this way.
Carrying Capacity (Logistic Growth)
The maximum population size of a biological species that can be sustained indefinitely by a given environment, represented by the horizontal asymptote in a logistic growth model.
- In the logistic function P(t) = K / (1 + A * e^(-rt)), K is the carrying capacity.
- As time (t) approaches infinity, e^(-rt) approaches 0, so P(t) approaches K.
- It signifies the upper limit of population growth in a constrained environment.
- Often seen in ecology and population dynamics.
Memory trick: Logistic levels off, the K is the ceiling, the maximum it can take.