A population of bacteria doubles in number every five minutes. If the population is one at three o’clock, what would the population be at four o’clock?
Well, this kind of problem is a problem dealing with exponential growth or an exponential formula. And what we can do is use this general form here to help us. So we know a function is equal to 𝐴𝑏 to the power of 𝑥. Well, 𝐴 is our initial value. 𝑏 tells us something about the rate. And it’s always a positive number that’s not equal to one. And then 𝑥 is our independent variable, which usually is the number of time periods.
So you might think, “Well, how do we know that this is an exponential problem?” Well, it’s because the population doubles every time period. So in this case, it’s every five minutes. It’s because of that doubling, which we know that it’s an exponential problem.
So what we’re gonna do first is we’re gonna use the information we’ve got to set up an equation. So we’re going to have 𝑝 as our function as it’s the population. It’s gonna be equal to one. And that’s because one is our initial value because the population of bacteria is one at three o’clock. And then we’re gonna have multiplied by two. And we get two because, as we said, the 𝑏 in our formula is something that tells us about the rate. Well, we can see that the bacteria doubles in number every five minutes so that’s why we have two.
And then we’re gonna have to the power of 12. And the reason we have that as our exponent is because if we look at the population, we’re trying to find out what the population of bacteria is going to be at four o’clock. Well, if we started at three o’clock, that means the time period is one hour. And we know that it doubles in number every five minutes. And there are 60 minutes in an hour. So we do 60 divided by five gives us 12. So there are 12 time periods. So that’s why I’ve got two the power of 12. So therefore, we can say that if a population of bacteria doubles in number every five minutes and the population was one at three o’clock, then the population at four o’clock is gonna be 4096.