Bobby's question at Yahoo! Questions regarding bacterial growth

[MarkFL]

Here is the question:

Differential Equation Problem?

A bacterium doubles in population every 6 hours. If the area that the bacterium is spread
over is proportional to its population (that is, if the population density remains constant),
and begins at 1 cm2, how long will it take the bacterium to fill the entire area of a Petri dish
of radius 2cm?

I have posted a link there to this thread so the OP can view my work.

 

[MarkFL]

Hello Bobby,

We don't need to solve an ODE, we can see from the given information that the population of the bacteria must be given by:

\(\displaystyle P(t)=P_02^{\frac{t}{6}}\)

Now, since the population density remains constant, we have:

\(\displaystyle A(t)=A_02^{\frac{t}{6}}\)

And we are told \(\displaystyle A_0=1\text{ cm}^2\) and so in square cm, we may write:

\(\displaystyle A(t)=2^{\frac{t}{6}}\)

Now, to find when the culture fills the dish, we may write:

\(\displaystyle 2^{\frac{t}{6}}=\pi(2)^2=4\pi\)

Taking the natural log of both sides, we obtain:

\(\displaystyle \frac{t}{6}\ln(2)=\ln(4\pi)\)

Solve for $t$:

\(\displaystyle t=\frac{4\ln(4\pi)}{\ln(2)}\approx21.9089767768339\)

Hence it will take about 21.9 hours for the culture to fill the dish.