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26 December, 00:58

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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  1. 26 December, 01:35
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    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
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