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Problem 68 Hard Difficulty

A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Show that the concentration of salt after $ t $ minutes (in grams per liter) is $$ C(t) = \frac{30t}{200 + t} $$

(b) What happens to the concentration as $ t \to \infty $?

Answer

$C(t)=\frac{30 t}{200+t}$

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Video Transcript

All right, so we start with 5000 liters of pure water. We're pumping in um uh salt concentration, we have 35 30 g of salt per leader. We're pumping that in at 25 liters per minute. And we want to find the salt Concentration of the whole thing after 10 minutes. So what happens? Um Well, first of all, let's think about the volume of the whole thing. We start with 5000 and after 10 minutes, uh we've added 25 liters per minute. So we're adding 25 T. And then how much salt is in there? We start with no salt. And then we're adding 30 grams per per leader But 25 Lpm. So we're adding 30 times 25 T rams. Um okay, and then uh if we just simplify that, let's divide top and bottom by 25. So we get 30 t. Over. This is going to be 200 plus T. Okay. And that they called that the concentration of salt in the water. Okay, now what happens as T. Goes to infinity? So we're looking for the limit as T goes to infinity of CFT just the limit, I should write a better see there as T. Goes to infinity of 30 T Over 200 plus T. And this is a big over big, right? They're both big. So I'm going to divide top and bottom by T. I think this is the most straightforward way to do this. So 32 divided by T is 30. This way things are small, not big. Small is a lot easier. Um 200 divided by T Plus T over T is one. And then the only theorem I'm using here is the limit as T gets big of one over T is zero, right? One divided by a big number is zero. And so I have that here, I have 200 times that. So that part is going to go to zero. Everything else is easy. I have 30/0 plus one. So um In the long term the concentration is going to be 30 and this makes sense. 5000 seems like a really big number, but they didn't tell you a cap on the volume. So for all, you know, you're filling up the ocean. And I started with uh 500 liters, 5000 liters of pure water. But eventually your concentration is going to be what you're pouring in if you're doing this forever. So that makes sense.