00:01
So this question is about an epidemic, and we're given an equation, which is that the rate of change of the number of people infected is some rate r, which is positive constant, times the number of people infected, times the total population minus the number of people infected.
00:20
Now, first of all, we're asked to identify a us state and give its population to the nearest million, and i've chosen new york state, which has a population of 20.
00:33
Million to the nearest million.
00:37
So now we're going to start off with the population of susceptible people is the total population.
00:45
We've already assumed that.
00:48
Now in part three, we're told that the initial condition for this equation is that the initial number of people infected is 1 ,000th of a percent, so 1 over 100 ,000 times the total population, which is is going to be, yeah, so it's the total population divided by 100 ,000.
01:13
So now we need to solve the equation.
01:18
So we've got df by d t equals r f p minus f, which we can integrate.
01:23
D f over f p minus f is equal to the integral of r d t.
01:30
And at t equals zero, f of zero is p over a hundred thousand, or ten thousand in fact, because it's one thousandth of a percent.
01:38
Oh no, that should be 100 ,000, yeah.
01:43
And then at t equals t, we have f of t.
01:46
So we need to integrate both sides of this equation.
01:48
On the t side, we can integrate just to get r t.
01:52
On this side, we're going to need to do partial fractions.
02:00
And so this is going to be some number over f plus some number over p minus f, integrated with respect to f.
02:11
And now when we put these back together, we're going to get pa plus f, b minus a.
02:20
Now there's no f in the denominator, so that in the numerator, so that tells us that b is equal to a.
02:26
And we also have that pa equals 1.
02:31
So a is 1 over p.
02:33
So this a is 1 over p, and this b is equal to a, so it's also 1 over p.
02:44
So actually what i'm going to do is i'm going to take the p's out of here and just put them in front of this, in front of this integral, because i can do that, 1 over p.
02:58
And now if i multiply both sides by p, i get p rt equals this integral here, which we can do.
03:07
Okay, so let's do it.
03:10
This is going to integrate to log f.
03:15
This is going to integrate to minus log p minus f, and we integrate between.
03:21
P over 100 ,000 and f of t.
03:27
But now the difference between two logs is the quotient of their argument.
03:33
So this is log f over p minus f, evaluated between p over 100 ,000 and f of t.
03:43
And now this is going to be log of f of t over p minus f of t minus lug of of p over 100 ,000 times p minus f so that's going to be 999999 p over 100 ,000 so the 100 ,000 is going to cancel and the p's are going to cancel and this minus log of 1 over 100999 remember minus log of something is log of 1 over that thing so we can just do a plus and put that here, so this becomes 999 .99.
04:34
But now we're adding two logs together, and that means we can multiply their arguments.
04:39
So we get that prt is equal to log of 99 ,999 f of t over p minus f of t.
04:55
And now we can exponentiate both sides.
04:58
We get that 99999 f of t, equals p minus f of t times e to the pr t.
05:10
So that tells us that f of t times 999 999 plus e to the p r t is equal to p r t.
05:26
Is equal to p e to the p r t...