# Doubling time is the time it takes a population to double at a constant rate of growth. Bacteria, for instance, multipy by division. One bacterium becomes two. Then two divide into four; the four...

Doubling time is the time it takes a population to double at a constant rate of growth. Bacteria, for instance, multipy by division. One bacterium becomes two. Then two divide into four; the four divide into eight and so on. For certain strains of bacteria, the time for this division process is one minute.

a. If you put one of these bacterium in a botle at 11:00 p.m., the entire bottle will be full by midnight. When would the bottle be half full?

b. How do you know?

c. suppose you could be a bacterium in this bottle. At what time would you first realize that you were running out of space?

d. Suppose that at 11:58 some bacteria realize that they are running out of space in the bottle. So they launch a search for new bottles. They look far and wide, and finally, offshore in the Arctic Ocean, they find three new empty bottles. Great sighs of relief come from all the bacteria. This is three times the number of bootles they've known. Surely, they think, their space problems are over. Is that so?

Since their space resources have quadrupled, how long can their growth continue.

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### 1 Answer

a. The bottle would be half full by 11:59.

b. By working backwards from the fact that capacity is reached at 12 Midnight, and the Doubling time is 1 minute. I know that the final doubling had to be from 1/2 full to completely full, and that put 1/2 full exactly 1 minute earlier.

c. Discounting Bacteria's lack of perception and comprehension, Sometime between 11:59 and 12:00 would be when I realize an impending crunch.

d. No their problems are not over. In just two minutes they will have the same problem. Because at 12:01, there will be enough bacteria to fill 2 bottles (previously there had been only enough for 1). At 12:02, that doubles to 4 bottles, and they are out of room again.