Digital SATMath: AlgebraHard
A scientist is tracking the population of a certain species of fish in a lake. The population, P(t), after t months, can be modeled by the equation P(t) = 1500 / (1 + 4e^(-0.2t)). What is the carrying capacity of the lake for this fish species?
- AInfinite
- B1900
- C1500
- D6000
Show answer & explanationAnswer & explanation
Correct answer: C. 1500
The given equation is a logistic growth model, P(t) = C / (1 + Ae^(-kt)). In this model, C represents the carrying capacity. As t approaches infinity, e^(-0.2t) approaches 0. Therefore, the denominator (1 + 4e^(-0.2t)) approaches 1 + 4(0) = 1. So, P(t) approaches 1500 / 1 = 1500. The carrying capacity is 1500.
Why the other options are wrong
- A. This is incorrect; logistic growth models indicate a limited carrying capacity, not infinite.
- B. This value might result from an incorrect interpretation of the constants in the denominator.
- D. This value might result from multiplying the numerator by a constant in the denominator, which is incorrect.
Carrying Capacity (Logistic Growth)
In a logistic growth model P(t) = C / (1 + Ae^(-kt)), the carrying capacity (C) is the maximum population size that the environment can sustain indefinitely.
- The carrying capacity is the upper limit of population growth.
- It is represented by the numerator (C) in the logistic function.
- As time (t) approaches infinity, the exponential term approaches zero, and P(t) approaches C.
Memory trick: Logistic Limit: The top number 'C' is the population ceiling!