00:01
So in order to do this, it's going to require us to obviously do some differential equations.
00:05
So let's first consider the pollutants in and the pollutants out because ultimately that's going to tell us the rate of change of the pollutants over time, which is pn minus p out.
00:22
So pn, we see is going to be 3 times 75, which is 225 pounds per hour, because we know that there are three pounds of pollutants entering with the water that's coming in at 75 gallons per hour.
00:43
And then we know that that water is being released from the tank at 50 gallons per hour.
00:47
So p out, this one's going to be a little longer, is going to be the amount of pollutant p divided by the initial amount of water, which is 200, plus 75 minus...
01:12
50t because we see that 75 is going in but 50 is going out so we see that overall 25 is going out every single hour and we multiply that by 50 because that is the rate at which the water is being released from the tank so with that we can simplify to be p out equals 2 y or 2 p over 8 plus t so now writing our dpd t together p n versus p out that ends up giving us 225 minus 2y over 8 plus t 2 p over 8 plus t so right there that's going to be um if we move this over here that's d p d t plus 2 p d p over 8 plus 2 p over 8 plus t equals 225.
02:21
So this is now a linear differential equation that we can solve, where p is 2 over 8 plus t, the other p now.
02:34
We're not talking about this p.
02:36
We're talking about the p for a linear equation.
02:39
And then q is 225.
02:43
So then what we're going to end up having is e to the integral of p -d -x, which equals e to the integral of 2 over 8 plus t d t, which equals e to the 2 natural log of 8 plus t, which equals e to the natural log of 8 plus t squared, which just equals 8 plus t squared.
03:23
Now the general solution, we end up getting to be p, the actual pollutants, times 8 plus t squared, equals 225 times the integral of 8 plus t squared d .t.
03:42
That's using our general solution for the linear differential equation...