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## Problem solving and data analysis

Current time:0:00Total duration:2:28

# Linear and exponential growth — Basic example

## Video transcript

- [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.