Digital SATMath: AlgebraMedium
A scientist is tracking the population of a certain type of algae in a pond. The population, P(t), in thousands, after t weeks, can be modeled by the function P(t) = 150 / (1 + 4e^(-0.5t)). What is the carrying capacity of the pond for this algae?
- A154 thousand
- B150 thousand
- C0.5 thousand
- D4 thousand
Show answer & explanationAnswer & explanation
Correct answer: B. 150 thousand
For a logistic growth model in the form P(t) = K / (1 + Ae^(-rt)), the carrying capacity is the value of K. In the given function, P(t) = 150 / (1 + 4e^(-0.5t)), K = 150. So, the carrying capacity is 150 thousand.
Why the other options are wrong
- A. This might result from adding 150 and 4, which is not how carrying capacity is determined.
- C. This is the 'r' value, which affects the growth rate, not the carrying capacity.
- D. This is the 'A' value, which relates to the initial population, not the carrying capacity.
Carrying Capacity (Logistic Growth)
In a logistic growth model, carrying capacity is the maximum population size of a biological species that can be sustained by the environment, represented by the numerator (K) of the logistic function.
- The population approaches the carrying capacity as time approaches infinity.
- Represented by the 'K' in P(t) = K / (1 + Ae^(-rt)).
- It's the upper limit that the environment can support.
Memory trick: K is the Ceiling, the top K-apacity.