The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 25% in

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The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 25% in 10 years. What will be the population in 40 years

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Alice 1 month 2021-10-14T01:31:29+00:00 1 Answer 0 views 0

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    2021-10-14T01:33:06+00:00

    Answer:

    The population in 40 years will be 1220.

    Step-by-step explanation:

    The population of a town grows at a rate proportional to the population present at time t.

    This means that:

    P(t) = P(0)e^{rt}

    In which P(t) is the population after t years, P(0) is the initial population and r is the growth rate.

    The initial population of 500 increases by 25% in 10 years.

    This means that P(0) = 500, P(10) = 1.25*500 = 625

    We apply this to the equation and find t.

    P(t) = P(0)e^{rt}

    625 = 500e^{10r}

    e^{10r} = \frac{625}{500}

    e^{10r} = 1.25

    Applying ln to both sides

    \ln{e^{10r}} = \ln{1.25}

    10r = \ln{1.25}

    r = \frac{\ln{1.25}}{10}

    r = 0.0223

    So

    P(t) = 500e^{0.0223t}

    What will be the population in 40 years

    This is P(40).

    P(t) = 500e^{0.0223t}

    P(40) = 500e^{0.0223*40} = 1220

    The population in 40 years will be 1220.

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