Digital SATMath: Advanced MathHard
A population of deer in a national park is modeled by the function P(t) = 1000 / (1 + 9e^(-0.2t)), where P(t) is the population after t years. What is the carrying capacity of the park for the deer population?
- A100 deer
- B1000 deer
- C9000 deer
- D200 deer
Show answer & explanationAnswer & explanation
Correct answer: B. 1000 deer
The carrying capacity of a logistic growth model, given by P(t) = K / (1 + Ae^(-rt)), is the value K. This represents the limit of the population as time (t) approaches infinity. As t → ∞, e^(-0.2t) → 0. So, P(t) → 1000 / (1 + 9 * 0) = 1000 / 1 = 1000. Therefore, the carrying capacity is 1000 deer.
Why the other options are wrong
- A. This might be a misinterpretation of a coefficient or a division error.
- C. This might be a result of multiplying by 9 or other misinterpretations of the denominator.
- D. This is incorrect; it's not directly derivable from the given logistic function parameters.
Carrying Capacity in Logistic Growth
In a logistic growth model, the carrying capacity (K) is the maximum population size of a species that the environment can sustain indefinitely, representing the upper limit or horizontal asymptote of the population function.
- Logistic model: P(t) = K / (1 + Ae^(-rt)).
- K is the maximum population the environment can support.
- As t approaches infinity, the population approaches K.
- The growth rate slows as the population approaches K.
Memory trick: Logistic is 'Limit'ed by the 'K'apacity 'K'ing.