Bobby's question at Yahoo Questions regarding bacterial growth

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In summary, the problem asks how long it will take for a bacterium to fill a Petri dish of radius 2cm, given that it doubles in population every 6 hours and the population density remains constant. Using the given information, we can determine that it will take approximately 21.9 hours for the culture to fill the dish.
  • #1
MarkFL
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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.
 
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  • #2
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.
 

FAQ: Bobby's question at Yahoo Questions regarding bacterial growth

What factors affect bacterial growth?

The growth of bacteria is affected by various factors such as temperature, pH, availability of nutrients, oxygen, and competition with other microorganisms. These factors can either promote or inhibit the growth of bacteria.

How can we control bacterial growth?

Bacterial growth can be controlled through various methods such as physical methods like heat treatment and filtration, chemical methods like using disinfectants and antibiotics, and biological methods like using beneficial microorganisms to compete with harmful bacteria.

What are the consequences of uncontrolled bacterial growth?

If bacterial growth is left uncontrolled, it can lead to various consequences such as food spoilage, foodborne illnesses, and infections in humans and animals. It can also cause damage to crops, plants, and other materials.

How can we measure bacterial growth?

Bacterial growth can be measured by counting the number of bacterial colonies on a culture plate, using a spectrophotometer to measure the turbidity of a liquid culture, or by measuring the production of metabolic byproducts such as CO2 or ATP.

How does bacterial growth contribute to the ecosystem?

Bacterial growth plays a crucial role in the ecosystem by decomposing organic matter, cycling nutrients, and aiding in the digestion and decomposition of food in animals. Bacteria also play a vital role in nitrogen fixation, making it available for plants to use for growth.

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