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) = 800/(1+4e^-0.31t)

Answer 1

To find the initial population, we evaluate

$P(t)=\frac{800}{1+4e^{-0.31t}}$

at t=0:

$\begin{gathered} P(0)=\frac{8_{}00}{1+4e^{-0.31\cdot0}}=\frac{8_{}00}{1+4e^0} \\ =\frac{8_{}00}{1+4}=\frac{800}{5}=160. \end{gathered}$

Therefore, the initial population was 160 individuals.

To find the population after 10 years, we evaluate the given function at t=10:

$\begin{gathered} P(10)=\frac{800}{1+4e^{-0.31\times10}}=\frac{800}{1+4e^{-3.1}} \\ \approx678. \end{gathered}$

Therefore, the population after 10 years is 678 individuals.

Answer:

The initial population was 160 individuals.

The population after 10 years is 678 individuals.