00:01
All right, so for this problem, we have a salt tank initially contains 60 liters of solution.
00:08
19 grams of salt are initially there.
00:11
It tells us that salt is poured into the tank at a rate of 11 grams per liter with a flow rate of 7 liters per minute and that the solution is pumped out of the tank at 5 liters per minute.
00:27
So we're using x to represent the grams of salt in the tank and t to represent time in minutes.
00:34
Now the first part of this question is wanting us to figure out, well, what's the volume of the tank at any given time? because you'll notice the rates coming in and out are going to be different.
00:45
So we'll start off with that.
00:48
We'll call it volume.
00:52
Should be equal to, well, we have our 60 liters to start.
00:57
Now what happens after every minute? well, you're going to gain seven liters and then you're going to lose five.
01:03
Or in other words, you'll have a net gain of two liters every single minute.
01:08
So you'll have two liters per minute, or in other words, 60 plus 2t.
01:14
Okay? so that's our first part.
01:17
Now the second part is to come up with a differential equation.
01:20
So we want to have an equation that represents amount of salt with time.
01:25
So we have dxdt.
01:27
And this is going to equal the rate of salt coming in minus the rate of salt going out.
01:37
Now, the most important thing for these problems are the units of these rates.
01:42
So the dxdt, you'll notice the units for x should be grams of salt, and the units for time are minutes.
01:52
So each of these rates here, the rate in and the rate out, also need to be written in grams per minute.
02:00
So the rate coming in, we're going to end up multiplying some rates together...