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15 June, 21:45

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

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  1. 15 June, 21:56
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    the initial population P₀ was 5938 people

    Step-by-step 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
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