When is the glass half-full?

Logic Level 1

Imagine you have a laboratory glass in which you collect bacteria that reproduces by multiplying itself once each second. At the beginning of the experiment, you have your first and only bacterium. After one second, you get two bacteria. After three seconds, you'l get four bacteria. After a lapse of four seconds, there are 8 bacteria and so on.

If your glass is full of bacteria after one minute, at what second is the glass half-full?

I know this problem will be a piece of cake for most (if not all) of the participants in this website, but I remember this problem from long ago and I still like to pose it to my friends. You don't need calculus to get the answer, but I also remember that my calculus teacher at that time during high-school was showing us integrals, and he solved the problem right away using an integral. Unfortunately, I don't have the skill or the knowledge I had then to show you the solution using calculus, but I know many of you will.

I hope you like this simple exercise.

30 59 15

2 solutions

Ignacio Curiel
Feb 11, 2019

It is very easy. If the number of bacteria doubles itself each second, you only need one second to go from half-full to completetly full. So, the glass will be half full one second before one minute, that is, at 59 seconds.

William Allen
Feb 12, 2019

$\text{Full contains } 2^{n} \text{ bacteria, so half full is } \frac{2^{n}}{2} = 2^{n-1} \text{ bacteria, where n is the number of seconds. } \\ \text{If it is full at 60 seconds then it will be half full at } 60-1 = 59 \text{ seconds }$

When is the glass half-full?

2 solutions

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