Digital SATMath: Advanced MathHard
A population of fish in a lake is modeled by the function P(t) = 1000 / (1 + 9e^(-0.5t)), where P(t) is the population after t years. What is the carrying capacity of the lake for this fish population?
- A100 fish
- B9000 fish
- C10,000 fish
- D1000 fish
Show answer & explanationAnswer & explanation
Correct answer: D. 1000 fish
The given function is a logistic growth model, P(t) = K / (1 + Ae^(-kt)). In this model, K represents the carrying capacity. By comparing the given function P(t) = 1000 / (1 + 9e^(-0.5t)) to the general form, we can see that K = 1000.
Why the other options are wrong
- A. This is an incorrect interpretation of the logistic growth model, possibly confusing K with A or another parameter.
- B. This might be a miscalculation involving the 'A' term (9) in the denominator.
- C. This is an incorrect interpretation of the logistic growth model, possibly assuming an initial population or another parameter.
Carrying Capacity in Logistic Growth
The carrying capacity (K) in a logistic growth model represents the maximum population size that the environment can sustain indefinitely, given the available resources. It is the upper limit of the population.
- In P(t) = K / (1 + Ae^(-kt)), K is the carrying capacity.
- The population approaches K as t approaches infinity.
- It represents the equilibrium point where population growth rate slows down and eventually stops.
Memory trick: The top number is the limit, the carrying capacity's summit!