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

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