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## George the Bacteria

Posted by Karl Sharman on March 20, 2014 – 3:57 am

I have a pet bacteria (George). He/She is one micron in diameter and he/she reproduces by dividing every minute into two bacteria, both called George. At 12:00 PM, I put George in a container. At precisely 1:00 PM, the container was full.

It would be too easy to ask “At what time was the container half full?”, so, how big was the container?

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1. 1. Wizard of Oz Said：

Hi Karl,

Hi Karl,

Welcome back from Darkest Africa, as it used to be called in my childhood.

As for the bacteria problem I don’t have the time to work on it, but doesn’t the optimum packing density of spheres in a container come into this? Also the shape of the container?

I assume that there is enough food in the container to sustain George and his/her descendants through their constant increment of binary powers.

2. 2. Karl Sharman Said：

I am not so evil as to let poor George starve to death, and yes, he is at the optimum growth temperature of 37 degrees.

On the size of the container – thats Trick of the Mind for you – the bacteria COULD be spherical, therefore, Volume = (number of bacteria)*(volume of each spherical bacterium)/0.7405 giving the optimum packing size for spheres. Carl Gauss came up with that.

But the Merriam-Webster disctionary definition does not say that the object has to be circular or spherical, but certainly infers it.

So, being generous, now that I am back in civilization, with electricity (I missed that the most!) and feeling somewhat benevolent, I shall accept answers for both spherical and cuboid bacterium….

3. 3. Karl Sharman Said：

Well, as you may have guessed, the container was half full at 12:59 PM. When the bacteria doubled in the next minute, the container became full. This is an example of exponential growth where the growth rate is a mathematical function that is proportional to the function’s current value

How big was the container?

The number of bacteria in the container is 2^n where n is the number of minutes elapsed. The full container has 2^60 (two to the sixtieth power), or approximately 1.153×10^18 bacteria.

Since one micron (one millionth of a meter) is 1×10^-4 centimeters, each bacterium occupies a volume of about 1×10^-12 cubic centimeters. Multiplying times the number of organisms, we conclude that the container had a capacity of 1.153×10^6 cc or 1,153 liters. The container was slightly bigger than one cubic meter.

However, Wizard of Oz rightly commented on the optimum packing of spheroid bacterium.
I stated that each bacterium occupies a volume of 1×10^-12 cubic centimeters, or (1 micron)3. This would imply that the bacteria are cubic. However the problem states that the bacteria could be spherical (1 micron in diameter). The more accurate volume occupied by each bacterium is 0.5236×10^-12 cubic centimeters (4/3 × pi × r^3). Further complicating the problem is the fact that spheres stacked upon one another leave a considerable amount of empty space between them. It is well documented that [in a close-packed arrangement] the ratio of volume occupied by the spheres to the total volume of 3-dimensional space is 0.7405. (Carl Gauss) Therefore, the container capacity would be more precisely calculated as 0.8152×10^6 cc or 815.2 liters.
Volume = ((number of bacteria)*(volume of each spherical bacterium))*0.7405

Strictly speaking, the problem does not state that the bacteria are “spherical”. The word “diameter” is defined by the Merriam-Webster Dictionary as 1) a chord passing through the center of a figure or body, 2) the length of a straight line through the center of an object. The definition does not say that the object has to be circular or spherical. However, other sources define “diameter” as a straight line segment passing through the center of a figure, especially of a circle or sphere, and terminating at the periphery.

So there you have it. The volume of the container is 1,153 liters for cubic bacteria, or 815 liters for round bacteria arranged in a close-packed arrangement.

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