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29 September, 03:25

A town has a population of 2093 people at time t=0. The population increases by 42 people a year. Enter a formula for the population, P, of the town as a function of year t.

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  1. 29 September, 06:14
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    P (t) = 2093e^ (42t).

    Step-by-step explanation:

    The population of this town can be modeled by the following differential equation

    dP/dt = Pr

    where r is the growth rate in people a year.

    We can solve this differential equation by the separation of variables method.

    dP/P = rdt

    Integrating both sides, we have:

    ln P = rt + P0

    where P0 is the initial population

    To isolate P, we do this:

    e^ (ln P) = e^ (rt + P0)

    P (t) = P0e^ (rt).

    We have that the population increases by 42 people a year, so r = 42. We also have that the population at time t = 0 is 2093 people, so P0 = 2093.

    So the formula for the population, P, of the town as a function of year t is P (t) = 2093e^ (42t).
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