The document discusses a population growth model where the population increases proportionally to the current population size over time. It is given that the initial population doubled in 5 years. The model is used to calculate how long it would take for the population to triple and quadruple from the initial size. It is determined that it would take 8.4 years for the population to triple, and 10.66 years for it to quadruple.
1. KATHERINE LORENA SILVA ALONSOCODIGO: 2073612METODOS NUMERICOS-534035-455295 <br />EJERCICIO PROPUESTO DE MODELO MATEMATICO<br />POPULATION GROWTH AND DECAY TIME<br />The population increased at a rate that is proportional to the number of people present at time t. if an initial population has doubled in five years.¿How many the populations take to triple and quadruple?<br />SOLUTION<br />Let P = p (t) the population at time t<br />Now solve the initial value problem, PVI<br />dPdt=kP , P0=P0<br />Pt=Cekt<br />Applying the initial condition<br />P0=P0e0=C<br />Pt=P0ekt<br />Is P5=2P0<br /> P5=2P0=P0=ek5<br />ln2=lnek5=5k<br />k=ln25=0.138<br />Pt=P0e0.138<br />Is P(t1)=3P0=P0e0.138t<br />t1 is the time that spend the population in be the triple <br />ln3=0.138t1<br />t1=ln30.138=8.4 years<br />Is Pt2=4P0<br />t2 is the time that spend the population in be the Quad<br />P(t2)=4P0=P0e0.183t<br />ln4=0.13t<br />t=ln40.13=10.66 years<br />