When will the bacteria fill the test kit? [closed]

Question

A test kit, is a gathering place for a colony of bacteria. This Colony, at 12 pm cell division has created a bacteria. Every minute, every bacteria is divided into two bacteria. 12: 43 (43 minutes later), half of the kit is full of bacteria. When kit will be filled of bacteria?

Answer 1

The canonical answer is 12:44 PM, because if all the cells in that half of the culture divide, they will take up two halves, or the whole culture. However, this answer is too simplistic and ignores crucial facts in population dynamics.

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A more realistic answer would be approximately 1:30 PM, around double the time that it took to get from one cell to half the culture.

The reason for this is that bacterial cultures grow logistically, not exponentially. At 12:44 PM, the bacteria would not take up the whole culture, because a lot of the existing bacteria would have died out due to lack of nutrients or accumulation of waste products. Assuming the bacteria did in fact follow a perfect logistic curve, this is how the calculation would go:

Let $n$ be the proportion of the culture already occupied by bacteria, a number from $0$ to $1$. Because the bacteria double every minute at the start, $n(t)$ where $t$ is the number of minutes since 12:43 PM has a differential equation of $\frac{dn}{dt} = \log 2 \times n(1-n)$, and the logistic function itself is represented as $$\frac{1}{1 + 2^{-t}}.$$

Therefore, at 12:44 PM, the proportion of the culture taken up by bacteria would be $\frac{1}{1 + 2^{-1}} = \frac{2}{3}$, about two-thirds of the culture instead of the whole thing as the problem believes it will.

Now comes the question of when we decide that the "whole culture" has been occupied by bacteria, since the bacteria are obviously not going to literally fill up the entire culture at once, but instead the population levels out so that the population stops growing. (In biology, this is known as the bacteria having entered the stationary phase.) I would peg that point at the point where $\frac{dn}{dt}$ is less than 1 cell.

Since the logistic curve is odd, this means that the answer is 1:26 PM, since the logistic curve has the property that its derivative at $t$ minutes (rate of change at 1:26 PM) is equal to its derivative at $-t$ minutes (rate of change at 12:00, when the culture was just one cell, and the overall rate of change in population was also just one cell). Since bacterial culture growth is not exact, there is a little bit of fudge factor and I could reasonably predict that the bacteria would fill the culture entirely at 1:30 PM.

Answer 2

12:44

Too easy though! I would have added some red herrings at least to make it look complicated if I were you. :)