## A population is growing at a rate proportional to its size. After four years the population was 171,000 after 19 years the population size 2

Question

A population is growing at a rate proportional to its size. After four years the population was 171,000 after 19 years the population size 247,000 what was the original population calculs

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2 weeks 2021-09-10T11:19:05+00:00 1 Answer 0

the initial population was 155027 people

Step-by-step explanation:

since the rate of increase of the population is proportional to the population itself, then the corresponding formula for a population P and time t is

dP/dt = k*P , where k= constant

dP/P = k*dt

∫dP/P =∫ k*dt

integrating between time t=0 (with P=P₀) and time t=t (with P=P)

ln (P/P₀)=k*t

if the population was P= 171000, at t=4 years then

ln (171000/P₀)=k* 4years

then the population is P=247000 att=19 years , thus

ln (247000/P₀)=k*19 years

diving both equations

ln (247000/P₀) / ln (171000/P₀) =19 years/ 4 years

ln (247000/P₀) = ln (171000/P₀)^(19/4)

P₀/247000 = (P₀/ 171000)^(19/4)

(171000)^(19/4) / 247000 = (P₀)^(19/4) / P₀ = P₀^(15/4)

P₀ = 171000^(19/15) /[247000^(4/15) ] = 155027 people

therefore the initial population was 155027 people