00:01
So in this problem, we're told that there was initially 8 ,000 bacteria, and then after two hours, there would be 12 ,400.
00:08
So in part a, we want to find the function, n of t equals n -0, e -raise -to -r -r -t power, to model the number of bacteria after t -hours.
00:18
Well, n -sub -zero represents the initial amount, which would be 8 ,000.
00:22
And the only thing that we would need to know is our r value.
00:25
So we have to solve this for r.
00:27
Well, because we know after 2 hours, there's 12 ,000 400, we can substitute 12 ,400 in for n of t, n sub zero, we already mentioned was the initial amount, which is 8 ,000.
00:38
And then we have e raised to the r times t, which is two hours.
00:43
So now, like i said, we're going to solve this for r.
00:45
So the first thing we're going to do is we're going to divide both thousands by 8 ,000.
00:50
So in this case, that's going to reduce a couple of those zeros.
00:53
So we're going to have 124 divided by 80 equal to e raised to the 2r power.
00:59
So because we're dealing with base e, i'm now going to take the natural log of both sides.
01:03
So we're going to have the natural log of 124 over 80 equal to the natural log of e raised to the 2r power.
01:10
And the reason why we do this is because the natural log of e to the x is simply just x, which means that the natural log of e to the 2r power is simply just 2r.
01:19
So now we're going to have the natural log of 124 divided by 80 equal to 2r.
01:25
So now we can solve for r by dividing both sides by 2.
01:29
So now we can go to our calculation.
01:31
So we're going to have the natural log of 124 divided by 80, and then we're going to divide that value by 2, which means that r would be approximately, i'm going to round two places after the decimal, 0 .22.
01:45
So that would be our r value.
01:48
Next, let's take a look at part b.
01:50
In part b, what they want us to figure out is the growth rate.
01:53
Remember, that would be our r value, but remember, we need to make it a percent.
01:56
So we remove the decimal point two places to the right, which means that our growth rate would be approximately 22%.
02:04
Now for part c, it says, oh, actually, let me go back to part a.
02:08
We solved for r, but i forgot to go back and actually write our function.
02:11
So our function would be n of t equals that initial amount, which is 8 ,000, times e raised to our r value, 0 .22 t power...