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).

[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]

[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]

[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]