Question

A species of animal is discovered on an island. Suppose that the population size ( P(t) ) of the species can be modeled by the following function, where time ( t ) is measured in years: ( P(t) = {600}/{1 + 8e^{-0.18t}} ). Find the initial population size of the species and the population size after ( t ) years. Round your answers to the nearest whole number as necessary.

a) Initial population: 600; Population after ( t ) years: ( {600}/{1 + 8e^{-0.18t}} )
b) Initial population: 0; Population after ( t ) years: 600
c) Initial population: 600; Population after ( t ) years: 600
d) Initial population: 600; Population after ( t ) years: 0

Answer 1

The initial population size of the species is 67 when rounded to the nearest whole number, and the population after t years can be found using the function P(t) = \frac{600}{1 + 8e^{-0.18t}}.

Step-by-step explanation:

To determine the initial population size and the population size after t years for a species, we can use the provided population model function P(t) = \frac{600}{1 + 8e^{-0.18t}}. To find the initial population, we substitute t = 0 into the function. The exponential term e^{-0.18t} becomes e^0, which is 1, and the population model simplifies to P(0) = \frac{600}{1 + 8}, which calculates to 600/9. After simplifying, we find that the initial population size is 66.67, which when rounded to the nearest whole number is 67.

To find the population size after t years, we use the given function directly. For any non-zero value of t, the value of the population P(t) will be a function of t and can be calculated by substituting the value of t into the model function.