Answer:
The population size at t = 1 is approximately 282.23.
Explanation:
The function P(t) is given by:
P(t) = 600 / (1 + 7e^(-0.31t))
This function represents a logistic growth model, which is commonly used in biology to describe the growth of populations. In this model, P(t) represents the population size at time t.
The logistic growth model has several key parameters, including the initial population size, the carrying capacity (i.e., the maximum population size that can be sustained), and the growth rate. In this case, the carrying capacity is assumed to be 600, since that is the value that P(t) approaches as t gets very large.
The growth rate is determined by the parameter -0.31 in the exponent of the denominator. Specifically, as t increases, the term e^(-0.31t) approaches zero, which means that the denominator approaches 1. As a result, P(t) approaches the carrying capacity of 600. The larger the absolute value of the growth rate parameter, the faster the population grows towards its carrying capacity.
To evaluate the function P(t) at a specific time t, we simply substitute the value of t into the function. For example, to find the population size at t = 1, we can write:
P(1) = 600 / (1 + 7e^(-0.31*1))
P(1) = 600 / (1 + 7e^(-0.31))
Using a calculator, we can evaluate this expression to get:
P(1) ≈ 282.23
Therefore, the population size at t = 1 is approximately 282.23.