A population of protozoa develops with a constant relative growth rate of 0.6671 per member per day. On day zero the population consists of

Question

A population of protozoa develops with a constant relative growth rate of 0.6671 per member per day. On day zero the population consists of 4 members. Find the population size after 7 days. Since the relative growth rate is 0.6671, then the differential equation that models this growth is

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Harper 2 weeks 2021-11-19T16:25:51+00:00 1 Answer 0 views 0

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    2021-11-19T16:27:10+00:00

    Answer:

    The population size after 7 days is about 427.

    Step-by-step explanation:

    If y(t) is the value of a quantity y at time t and the if the rate of change of y with respect to t is proportional to its size y(t) at any time, then

                                                      \frac{dy}{dt} =ky

    where k is a constant.

    This equation is sometimes called the law of natural growth (if k>0).

    The only solutions of the differential equation \frac{dy}{dt} =ky are the exponential functions

                                                      y(t)=y(0)e^{kt}

    Let P be the population size and let t be the time variable, measured in hours. Since the relative growth rate is 0.6671, then the differential equation that models this growth is

                                                     \frac{dP}{dt} =0.6671\cdot P

    According with the above information the solution to this differential equation is

                                                    P(t)=P(0)e^{0.6671t}

    On day zero the population consists of 4 members P(0)=4.

    Therefore, the population size after 7 days is

    P(7)=4e^{0.6671\cdot 7}\\\\P(7)=4e^{4.6697}=4\cdot \:106.66573=426.66295

    about 427.

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