Digital SATMath: Advanced MathMedium
A biologist is studying a population of insects that grows according to the function P(t) = 1500 / (1 + 4e^(-0.2t)), where P(t) is the population after t days. What is the carrying capacity of this environment for the insect population?
- A300
- B1500
- C1900
- D4
Show answer & explanationAnswer & explanation
Correct answer: B. 1500
For a logistic growth model in the form P(t) = C / (1 + Ae^(-kt)), the carrying capacity is represented by the constant C (the numerator). In this function, P(t) = 1500 / (1 + 4e^(-0.2t)), the numerator is 1500, which is the carrying capacity.
Why the other options are wrong
- A. This might be a calculation error or a misinterpretation of the initial value.
- C. This might be 1500 + 400 or another incorrect sum.
- D. This is the constant 'A' in the denominator, related to the initial population, not the carrying capacity.
Carrying Capacity in Logistic Growth
In a logistic growth model of the form P(t) = C / (1 + Ae^(-kt)), the carrying capacity (or limiting value) is the maximum population that the environment can sustain indefinitely, represented by the constant C.
- The population approaches C as t approaches infinity.
- C is the numerator of the logistic function.
- It represents the upper bound of the population.
Memory trick: Carrying Capacity is the Ceiling, Top Number is King!