Linear and exponential growth — Basic example

- [Instructor] The following equation represents the population growth of bacteria in a petri dish. So, the population as a function of time is equal to 350 times two to the t power, where P of t is the population after some amount of time, t, measured in hours. Which of the following best describes the relationship between the population of bacteria, P, which is a function of t, and the number of hours that have passed? And the number of hours is that lowercase t. The relationship is linear since the population has 350 more bacteria than the previous hour. Well, this is clearly, or we'll have to see this is not a linear relationship right over here. What's gonna happen here? When t equals zero, you could even see it if you draw a little table here. So, if you say t and P of t, when t is zero, well then the population, two to the zero power is one, one times 350 is, the population's going to be 350. When t is one, what's the population now? Well, two to the first power is two, times 350, it's going to be 700. When t is two, what's gonna be the population? Well, it's two to the second power is four. Four times 350 is gonna be 1,200, plus 200 is gonna be 1,400. So, what you see, what's happening, after after every hour, the population is doubling. And you see that right over here 'cause for every hour you're gonna take two to that power as you increment, and then you're gonna multiply that times 350. So, every hour, your population is going to double. After three hours, the population is gonna be 2,800. So, this is definitely not a linear relationship. You definitely have much more than just 350 more bacteria than the previous hour, so rule that out. The relationship is linear; no, I don't even have to finish reading that. The relationship is exponential, yep, that's what it looks like, since the population is 350 times larger than the previous hour. No, no, no, it's not 350 times larger, it's two times larger. So, let's, to get 350 times larger, it would be 350 that you would be raising to the exponent. So, we'll rule that one out. So, this better be our answer, but let's make sure it makes sense. The relationship is exponential since the population is two times larger, the population is two times larger than the previous hour. That's exactly what we found.