00:01
For this problem, we are examining population growth.
00:05
We were given that in a certain culture of bacteria, the number of bacteria increased sixfold in 10 hours.
00:13
So the question is, how long did it take the population to double? so here on the screen, i have some information that might help us.
00:22
It is the exponential growth function, and i wrote down what the variables stand for.
00:28
We're going to fill it in, try to figure out.
00:31
What is the equation or function for this particular culture of bacteria? and then we'll use it to answer our question.
00:40
So let's get started here.
00:42
We were given that for this particular culture of bacteria, the number of bacteria increased sixfold in 10 hours.
00:50
So the population at time t, and i'm going to write here time is in hours, just to remind yourselves.
00:59
So when t is 10 hours, our population is going to be six times of what it started.
01:06
We don't actually know what was the initial amount.
01:10
So what we're going to do is for p, we're going to write six times of the initial population.
01:17
So six p not.
01:19
We call it p.
01:20
Not.
01:22
So this is six p .0 equals to p.
01:25
Not.
01:25
Again, we don't know what the initial population is, so we'll just leave it there.
01:30
E, we don't know what the rate is.
01:32
We're going to have to figure that out.
01:34
And t is going to be 10 for 10 hours.
01:38
So times 10.
01:40
Now the reason we don't really need to know what the initial population is is if you take a look at this equation, if i divide both sides by p .0 or p .0, it goes away.
01:54
So we don't really have to worry about it very much.
01:58
What we have to do right now is solve for so our rate.
02:05
So to do this, since we have an e here, we're going to do natural log of both sides of the equation.
02:16
And according to a log rule, when we have a log written like this, we can take the power and move it to the front of the log.
02:26
So on the right hand side of the equation, we'll have 10r times natural log of e.
02:35
Natural log of e is just one.
02:39
So don't need to really worry about it too much.
02:42
We're trying to solve for r.
02:44
This is 10 times r...