00:01
So in this question we're thinking about a health clinic, and we're told that the fraction still receiving treatment at the clinic after t months is s of t is equal to e to minus t over 36.
00:17
And this means that if we have n of zero equals n n -t, then n of t is going to be n -0, e to the minus t over 36.
00:34
So that means that the rate of change of the number of people, d n of t by d t, is equal to minus 1 over 36, n n, e to minus t over 36.
00:49
But n n n a to minus t over 36 is n of t.
00:53
So we get d n of t by d t is minus 1 over 36 n of t.
01:01
And this is describing how patients are leaving the clinic.
01:09
So patients leaving the clinic.
01:16
But we also have another contribution to the rate of change of the number, which is r of t is 20 patients per month, which is the patients entering the clinic.
01:40
So that means that we need to include this in our expression for how the rate, how the number of patients in the clinic is changing.
01:54
So dn by d t is 20 minus 1 over 36 times n.
02:03
So this represents how the number of patients in the clinic is changing.
02:08
And we also have an initial condition, which is that the initial number of patients in the clinic in the clinic, is 400.
02:16
So this is our first order initial value problem.
02:20
So first order initial value problem, which we need to solve.
02:27
And we're asked how many people will be in the clinic in 18 months.
02:31
So what is n of 18? okay, so let's work it out.
02:41
We need to integrate this equation.
02:42
So what we can do is we can write that the integral dn, we're separating this equation.
02:48
So we divide by this side, 20 minus n over 36, is equal to the integral d t.
02:57
Now when t is zero, we have 400 patients, and we want to find out n of t, which we can do this way.
03:08
Now the right -hand side we can integrate is just t.
03:12
The left -hand side, we're going to get something like log -20 minus -t.
03:17
N over 36.
03:20
When we take a derivative of this, we'll get the 1 over 20 minus n over 36, but we will also get a minus 1 over 36.
03:28
So we need to write minus 36 here.
03:32
And we evaluate this between 400 and n of t...