00:01
So in this problem, we're told that a bacteria culture starts with 400 bacteria, and that after four hours, there are now 25 ,600.
00:07
So we want to find the function in this form, n of t, equals n sub 0 times e raised to the rt power to represent the number of bacteria after t hours.
00:17
Well, remember, n sub zero is the initial amount.
00:20
So all we need is the initial amount, which we have, and we need our r.
00:24
So in this case, we don't know r, so that's what we need to solve for.
00:27
But because we know after four hours, there is 25 ,000, 6 ,000.
00:30
That would be our n of t value.
00:32
So we'll have 25 ,600 equal to n sub zero, which is 400, times our base e raised to the r times four power, because it took four hours.
00:42
So now we just have to solve for r.
00:44
So i'll begin by dividing both sides of our equation by 400.
00:48
Notice, those zeros at the end will cancel.
00:50
So we have 256 divided by four, which is 64, equal to to e raised to the four r power.
00:56
Now, because we're dealing with base e, i'm going to now take the natural log of both sides of this equation.
01:02
I do this because one of our properties for the natural logs says the natural log of e raised to the x power is just x, the exponent.
01:10
So on the right -hand side, the natural log of e to the 4 -r power is just 4r.
01:14
So we'll have the natural log of 64 equal to 4r.
01:18
So to solve for r, we just divide both sides of our equation by 4.
01:22
So now that with isolated r, we can go ahead and solve.
01:25
So we're going to have the natural log of 64 getting divided by 4.
01:30
I don't know if they want you to actually go ahead and round that, but i'll just leave it like this...