ACT (Enhanced)MathematicsHard
A research team is studying the population growth of a rare species. The population (P) after 't' years is modeled by the function P(t) = 500 / (1 + 4e^(-0.1t)). What is the carrying capacity (maximum population) of this species?
- A0.1
- B2000
- C4
- D500
Show answer & explanationAnswer & explanation
Correct answer: D. 500
This is a logistic growth model. The carrying capacity is the upper limit that the population approaches as time (t) approaches infinity. As t → ∞, e^(-0.1t) → 0. Therefore, the denominator (1 + 4e^(-0.1t)) → (1 + 4*0) = 1. So, P(t) → 500 / 1 = 500. The carrying capacity is 500.
Why the other options are wrong
- A. This is the 'b' coefficient in the exponent, which affects the rate of growth but not carrying capacity.
- B. This might be 500 * 4, which is not relevant to the carrying capacity of this model.
- C. This is the 'a' coefficient in the denominator, which affects initial growth but not carrying capacity.
Logistic Growth Model (Carrying Capacity)
A mathematical model describing population growth that levels off as it approaches a maximum limit (carrying capacity) due to environmental constraints.
- Formula: P(t) = K / (1 + ae^(-bt)).
- K is the carrying capacity.
- As t → ∞, P(t) → K.
Memory trick: Logistic Limit: As time goes infinite, the top number is the cap!