ACT (Enhanced)MathematicsHard
A biologist is tracking the population of a certain species of fish in a lake. The population (P) after t years can be modeled by the logistic growth function P(t) = 1200 / (1 + 3e^(-0.5t)). What is the carrying capacity of the lake for this fish species?
- A2400 fish
- B1600 fish
- C3 fish
- D1200 fish
Show answer & explanationAnswer & explanation
Correct answer: D. 1200 fish
In a logistic growth model of the form P(t) = K / (1 + Ce^(-rt)), K represents the carrying capacity. By comparing the given function P(t) = 1200 / (1 + 3e^(-0.5t)) to the general form, we can see that K = 1200. The carrying capacity is the maximum population the environment can sustain, which is the limit of P(t) as t approaches infinity.
Why the other options are wrong
- A. This is a plausible distractor, perhaps from multiplying K by C or similar error.
- B. This is a plausible distractor, perhaps from misinterpreting parts of the formula.
- C. This is the constant 'C' in the denominator, not the carrying capacity.
Logistic Growth Carrying Capacity
In a logistic growth model P(t) = K / (1 + Ce^(-rt)), the carrying capacity (K) is the maximum population size that the environment can sustain indefinitely, representing the upper limit of growth.
- K is the numerator of the logistic function.
- Population approaches K as time (t) approaches infinity.
- Growth rate slows as population approaches K.
Memory trick: K is for 'K-apacity', the top limit.