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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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Answer:

[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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