Digital SATMath: Advanced MathMedium
A biologist is studying a population of insects that grows according to the function P(t) = 120 / (1 + 19e^(-0.4t)), where P(t) is the population after t weeks. What is the carrying capacity of this insect population?
- A120
- B0.4
- C19
- D20
Show answer & explanationAnswer & explanation
Correct answer: A. 120
For a logistic growth model in the form P(t) = K / (1 + Ae^(-rt)), the carrying capacity is represented by the constant K in the numerator. In this function, K is 120.
Why the other options are wrong
- B. This is the growth rate constant 'r' in the exponent.
- C. This is the constant 'A' in the denominator, related to the initial population, not the carrying capacity.
- D. This value does not directly correspond to any parameter in the standard logistic growth formula.
Carrying Capacity in Logistic Growth
The maximum population size of a biological species that can be sustained indefinitely by a given environment, represented by the constant K in the numerator of the logistic growth function.
- Logistic growth models show initially exponential growth, then slowdown as population approaches carrying capacity.
- The carrying capacity is the horizontal asymptote of the logistic curve.
- It represents the upper limit of the population.
Memory trick: The 'K' on top is where life will stop!