Question

The population of a community is known to increase at a rate proportional to the number of people present at time t. The initial population P0 has doubled in 5 years. Suppose it is known that the population is 9,000 after 3 years. What was the initial population P0

Answer 1

the initial population P₀ was 5938 people

Explanation:

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

dP/dt = k*P , where k= proportionality 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 doubled , so P=2*P₀ at t=5 years then

ln (2*P₀/P₀)=k*5 years

k = ln 2 / 5 years

then the population is P=9000 for t= 3 years

ln (9000/P₀)=k*3 years

ln (9000/P₀)=ln 2 / 5 years *3 years

solving for P₀

P₀ = 9000 people * e^ (- ln 2 / 5 years *3 years ) = 9000 people *2 ^(-3 years / 5 years) = 5937.78 people ≈ 5938 people

therefore the initial population P₀ was 5938 people