The growth rate of escherichia coli, a common bacterium found in the human intestine, is proportional

The growth rate of escherichia coli, a common bacterium found in the human intestine, is proportional to its size. under ideal laboratory conditions, when this bacterium is grown in a nutrient broth medium, the number of cells in a culture doubles approximately every 15 min. (a) if the initial population is 500, determine the function q(t) that expresses the growth of the number of cells of this bacterium as a function of time t (in minutes).

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  1. [tex]Q(t) =[/tex][tex]100(2)^t[/tex]

    Step-by-step explanation:

    please see the attached files for more details

    [tex]Growth of Bacteria The growth rate of Escherichia coli, a common bacterium found in the human intest[/tex]
    [tex]Growth of Bacteria The growth rate of Escherichia coli, a common bacterium found in the human intest[/tex]

  2. [tex]Q(t) = 100e^{0.0231t}[/tex]

    Step-by-step explanation:

    The equation for the number of cells after t minutes is given by the following formula:

    [tex]Q(t) = Q(0)e^{rt}[/tex]

    In which Q(0) is the initial population and r is the growth rate.

    Initial population of 100

    So [tex]Q(0) = 100[/tex]

    Doubles after 30 minutes.

    So Q(30) = 200.

    We use this to find r

    [tex]Q(t) = Q(0)e^{rt}[/tex]

    [tex]Q(t) = 100e^{rt}[/tex]

    [tex]200 = 100e^{30r}[/tex]

    [tex]e^{30r} = 2[/tex]

    [tex]\ln{e^{30r}} = \ln{2}[/tex]

    [tex]30r = \ln{2}[/tex]

    [tex]r = \frac{\ln{2}}{30}[/tex]

    [tex]r = 0.0231[/tex]

    So

    [tex]Q(t) = 100e^{0.0231t}[/tex]

  3. [tex]N_{t} = 500 *e^{0.0462*t}[/tex]

    Explanation:

    The growth of Escherichia coli, in ideal conditions as described, follows an exponential curve, that can be aproximated by the following equation:

    [tex]N_{t} = N_{0} *e^{r*t}[/tex]

    because we know the doubling time, which is equal to 15 minutes, we can rearrange the equation, to find the constant r:

    [tex]2N = N*e^{r*t}\\2=e^{r*t}\\Ln 2=r*t\\\frac{Ln 2}{15 min} = r[/tex]

    r=0.0462

    Finally we reemplace the values in the equation

    [tex]N_{t} = 500 *e^{0.0462*t}[/tex]

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