Question

Exponential Functions in Action (5 points) To model population growth, researchers create the logistic growth function of the form РОК P(t) = P + (K - Pert where t is time, P, >0, K > 0 and r > 0. r is the base growth rate. (a) Find P(0) and lim P(t). Based on these answers, how can you interpret K and Po?

Answer 1

P(0) is equal to the carrying capacity K. The limit of P(t) as t approaches infinity is P. K represents the carrying capacity of the population and Po represents the initial population size.

Step-by-step explanation:

To find P(0), we can substitute t=0 into the logistic growth function P(t) = P + (K - P)e^(-rt). This gives us P(0) = P + (K - P)e^(0), which simplifies to P(0) = P + (K - P) = K. Therefore, P(0) is equal to the carrying capacity K.

To find the limit of P(t) as t approaches infinity, we can observe that as t gets large, e^(-rt) approaches 0. This means that the second term of the equation (K - P)e^(-rt) approaches 0, and P(t) approaches P + 0 = P. Therefore, the limit of P(t) as t approaches infinity is P, which represents the carrying capacity.

Based on these answers, we can interpret K as the carrying capacity of the population, which is the maximum population size that a particular environment can sustain. Po represents the initial population size at time t=0.