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)=\frac{790}{1 + 2e^{-0.18t}})find the initial population size of the species and the population size after 8 years. round your answers to the nearest whole number as necessary.initial population size: individualspopulation size after 8 years: individuals

Explanation:

Step1: Find initial population size

Set \(t = 0\) in \(P(t)=\frac{790}{1 + 2e^{-0.18t}}\). Since \(e^{-0.18\times0}=e^{0}=1\), we have \(P(0)=\frac{790}{1 + 2\times1}\).
\[P(0)=\frac{790}{3}\approx263\]

Step2: Find population size after 8 years

Set \(t = 8\) in \(P(t)=\frac{790}{1 + 2e^{-0.18t}}\). First, calculate \(e^{-0.18\times8}=e^{-1.44}\approx0.2369\). Then \(1+2e^{-1.44}=1 + 2\times0.2369=1 + 0.4738 = 1.4738\). So \(P(8)=\frac{790}{1.4738}\approx536\)

Answer:

Initial population size: 263 individuals
Population size after 8 years: 536 individuals