In this video, we're going to talk about using generation times to calculate the number of cells. And so if you are given the generation time for a particular microbial population, then the following equation, which you can see down below right here, can be used to calculate the number of cells after a certain period of time. And so notice that in this equation, there are several different variables that we define down below.

The first of these variables is \( n_t \). And \( n_t \) is equal to the final number of cells after a given amount of time. Then we have \( n_0 \) here. And \( n_0 \) is equal to the initial number of cells at the very beginning. And this is going to be multiplied by \( 2 \) raised to the power of \( n \), where \( n \) is an exponent here. And \( n \) is equal to the number of new generations over a given amount of time. And you can get \( n \) by taking the given amount of time and dividing it by the generation time, which should be given to you.

To show you how this equation here can be applied, we have an example down below. Notice that this example says to calculate the number of cells after 3 hours of growth, starting from 10 cells with a generation time of 30 minutes. Once again, we want to use this equation that's up above.

So, we'll start with the very first variable here, \( n_t \). \( n_t \), once again, is the final number of cells after a given amount of time. And notice that this is really what we want to calculate. We want to calculate the final number of cells after 3 hours of growth. So, \( n_t \) is our missing variable that we need to solve for. \( n_t \) is going to be equal to \( n_0 \cdot 2^n \). But \( n_0 \), again, is the initial number of cells. Notice that in the problem, it tells us that we are starting from 10 cells.

So, \( n_0 \) is going to be 10. Then we want to multiply \( n_0 \) by \( 2 \) raised to the power of \( n \). \( n \), again, is going to be the number of new generations over a given amount of time. To calculate \( n \), we need to convert the given amount of time, which is 3 hours, into minutes so that the time units match each other. The generation time is given in minutes, and the given amount of time is in hours. So 3 hours is a total of 180 minutes. \( n \) is going to be equal to the given amount of time, 180 minutes, divided by the generation time of 30 minutes.

When you calculate \( 180 \text{ minutes} \div 30 \text{ minutes} \), you'll get that \( n \) is equal to 6. So, \( n_t \) is equal to \( 10 \cdot 2^6 \). If you type that into your calculator, \( 2^6 \) comes out to 64. Then, if you take 64 and multiply it by 10, you get \( n_t \) equal to 640.

So, \( n_t \) equaling to 640 matches with answer option d. We can mark that d is the correct answer for this example problem. This equation can be used when you are given certain variables, and by applying this equation, you can calculate the number of cells after a given amount of time. We'll be able to get some practice applying this equation as we move forward. So I'll see you all in our next video.