 Imagine we have a large mixing tank currently containing 100 gallons of water into which five pounds of sugar have been mixed. So maybe we're making like a giant, a giant bowl of Kool-Aid, although you might want more sugar than that for your Kool-Aid. Anyways, a tap will open pouring 10 gallons per minute of water into the tank at the same time sugar is being pulled and poured into the tank at a rate of one pound per minute. Let's find the concentration that is pounds per gallon of sugar in the tank after 12 minutes. Is that a greater concentration than at the beginning? So imagine, let's kind of draw a picture to illustrate what's going on here. We have this big tank of water and this tank of water, it initially had 100 gallons of water in it. So this, if we think of like some time situation here when time equals zero, we have some initial volume, we'll call it V zero, which is 100 gallons of water. And then we had some initial salt or sugar, that's what it was. So we'll call that one say why not right here. And so we started off with five pounds of sugar starting this thing off and the volume was 100 gallons. And so I want you to kind of think of this for a second, this gives us a concentration. So the concentration of our sugar at the beginning, it looks like, well, this is just going to be our initial concentration, C not here. This is supposed to be why not divided by V not, which would then give us five pounds of sugar per 100 gallons of water. And since we're dividing by zero or divided by 100, we can just move the decimal over by two places. So we have this 5% concentration basically, which really we should think of it as just units. We have 0.05 pounds per gallon. So that's not a super high concentration. This 5% concentration is going on here. So that's going to help us answer the later question just initially. But now the thing is changing. The amount of water is changing into this system here. So what we're doing is we're adding 10 gallons per minute. So you're going to add 10 gallons per minute. So with respect to time, this is changing. We're also adding sugar. So we're adding sugar so that we're increasing it by one, one pound per minute in terms of sugar. And so we want to see what happens 12 minutes later. So what I want to do is first find a model for the volume and for the sugar. Now since we're adding 10 gallons per minute, this right here describes to me a rate of change. This is a rate of change. And in fact, we have this constant rate of change, which indicates to me we have a linear function. It's linear. We have a linear function for volume. So our volume is going to look like, our volume here is going to look like some MT plus some initial value B, right, where M is the slope, the rate of change that's constant and some initial value, which we're increasing the volume by 10 gallons per minute. And we started off with 100 gallons. So our volume function will be given as 100 T plus, excuse me, 10 T plus 100. The sugar concentration will be very similar. Y equals, we're increasing it by one pound per minute. So T plus five. So we get Y equals T plus five. And so when we put this together, our concentration function, it's supposed to look like Y divided by volumes, the amount of sugar divided by volume. We see that the concentration is going to equal T plus five over 10 T plus 100. And so we see that this right here gives us a rational expression. The concentration will be a rational function going on here, which you could factor the denominator. There's a common factor of 10 there. Pull out the 10. So you get 10 times 10 plus T plus 10, which, you know, we could say something about vertical asymptotes, but that's outside the domain of this problem, right? We should mention that T has to be greater than or equal to zero before the experiment started. Nothing makes any sense. We're not going to go back in time like Marty McFly or anything like that. So what was asked though was what is the concentration after 15 minutes, after 12 minutes? What happens there? So we have this initial 5% value. We want to compare that to what's the concentration after 12 minutes. So we're going to get 12 plus five over 10 times 12 plus 10, for which we get 17 on top 10 times, we're going to get 22 right there. So we get 17 over 220. That's our concentration. This is going to be pounds per gallon, which we should simplify this thing. And we end up with, well, as a decimal, this is going to be approximately 0.077 pounds per gallon. And so that's the comparison we want to make. So when we're at zero minutes, we start off with this 5% concentration. And then at 12 minutes, we're now at a 7.7% concentration. So that's how the concentration has changed over time. It definitely is going up because we're adding sugar to the system. But what you have to be careful about that is that the reason it's going up is that the concentration that we're inserting is bigger than we started with. So one pound per 10 gallons. That's actually a 10% concentration that we're adding to a 5% concentration. So this is going to go up and up and up and up and up. And you can see this, in fact, if we think of the in behavior of this function, if we ask what happens as T goes to infinity, if we allow this process to continue for forever, looking at the leading terms there, this thing, we see that the concentration will approach 1 over 10. That is one tenth or 10%. So this is the concentration that we're inserting into the system. And if we allow the system to continue on and on and on, the concentration will tend towards 10%.