00:03
Bacteria from a lab culture grows in such a way that the instantaneous rate of change of the bacteria, which is the change in bacteria over time, is directly proportional to the number of bacteria present.
00:17
So that would be a differential equation that expresses the relationship.
00:23
And you want to solve this for the number of bacteria as a function of time.
00:28
So first i would multiply both sides by the dt.
00:37
And then divide both sides by b.
00:44
And then you're able to integrate both sides.
00:48
So 1 over b is the natural log absolute value of b.
00:52
And then you're integrating the k with respect to t.
00:55
And we put the plus c on that side.
00:59
Now to solve for b, you're going to do e on both sides and let each side be the exponent for e.
01:06
That'll allow you to use the fact that e.
01:12
And natural logger inverses, and those two cancel, and you just have b, and at this point, you assume the b is positive.
01:20
And on the other side, you're going to use this fact that when you multiply things with the same base, you add the exponent.
01:29
So i'm just sort of like doing that in reverse.
01:32
And then e is a number, c is a number.
01:36
A number raised to a number is a bigger number.
01:39
And that gives you the equation for the number of bacteria as a function of time.
01:46
So that's your a answer.
01:49
For your b answer, initially there are 200 bacteria presence.
01:55
That means at times zero, bacteria is 250.
02:01
And then three hours later, bacteria grows to 625.
02:08
On the bacteria population after one day.
02:11
Okay, these two pieces of information is going to help you get the equation.
02:14
So we're going to plug in this one, the b and the time zero.
02:20
So 250 equals c, if t is zero, then the exponent is zero.
02:30
E to the zero is one...