Digital SATMath: Advanced MathMedium
A scientist is observing the population growth of a certain bacteria culture. The population P(t) after t hours is modeled by P(t) = 1000 / (1 + 9e^(-0.5t)). What is the carrying capacity of this bacteria culture?
- A500 bacteria
- B9000 bacteria
- C1000 bacteria
- D100 bacteria
Show answer & explanationAnswer & explanation
Correct answer: C. 1000 bacteria
The carrying capacity of a logistic growth model in the form P(t) = K / (1 + Ce^(-rt)) is given by the constant K in the numerator. In this function, P(t) = 1000 / (1 + 9e^(-0.5t)), the value of K is 1000. This represents the maximum population the environment can sustain.
Why the other options are wrong
- A. Incorrectly identifying the carrying capacity; 500 is not directly from the formula.
- B. This might come from an incorrect calculation involving the constant C (9) in the denominator.
- D. Incorrectly identifying the carrying capacity; 100 is not directly from the formula.
Carrying Capacity in Logistic Growth
The maximum population size of a biological species that can be sustained indefinitely by the environment, represented by the constant K in the logistic growth model P(t) = K / (1 + Ce^(-rt)).
- It is the horizontal asymptote of the logistic curve.
- Represents the upper limit of population growth.
- Found as the numerator 'K' in the standard logistic equation.
Memory trick: Logistic lid, the K is hid.