for-the-following-problems-set-up-and-solve-the-differential-equations-a-1000-liter-tank-contains-pu

Question

For the following problems, set up and solve the differential equations.
A 1000 -liter tank contains pure water and a solution of 0.2 kg saltL is pumped into the tank at a rate of 1 $\mathrm{L} /$ min and is drained at the same rate. Solve for total amount of salt in the tank at time $t$ .

Taylor Shimono · Accepted Answer

So we&#x27;re going to start this problem by setting up a formula for the rate of change of the amount of salt in the tank with respect to time. So we know that this race is going to be given to us by the rate of salt going in minus the rate of salt going out. So that&#x27;s what you tell us that the S d T will be equal to. We know that one of the solutions in Compton is going to a pump insult at a rate of 0.2 killer grounds for leader times 20 liters per minute. Well, give us kilograms per minute. Plus, our other solution is pumped in iterated 0.5 kilograms for leader times five liters per minute will again give us units of leaders per minute or accuse me kilograms for a minute in minus. Uh, the salt out will be. We know we&#x27;re pumping it out at a rate of 25 liters per minute. But the concentration of the assault that&#x27;s being pumped out is going to be equal to the amount of salt in the tank. I&#x27;m divided by the volume of tanks and so when we simplify this, Um, we can actually go ahead and substitute in, um, our salt the amount of salt in the tank is equal to the concentration times the volume because that&#x27;s going to give us units of kilograms. So the rate of change of salt in the tank with respect to time, um, is going to be equal to 0.0 4 to 5, um, minus, uh, 25 times concentration times the volume over the volume so we can see that this is going to cancel. And we know that the rate of change of the concentration with time is going to be equal to the amount of salt divided by the volume. So essentially, we&#x27;re just going to divide every term here by the volume. So when we do that, we get the CD T is equal to 0.0 for 4 to 5. Should have been not for five to minus 25 times a concentration over the volume. And, um, if we are to calculate the rate of change of volume with respect to time, we&#x27;re going to find that the volume is actually a constant 1000 leaders. You can calculate this on your own if you want to check me. But I&#x27;ve already calculated it. Um And so essentially, we&#x27;re just going to plug in 1000 for V. So when we do that, um, we get DC DT is equal to I&#x27;m in this step, actually already divided by 1000. I&#x27;m sorry if your number was a little bit off there, That would explain why that was my bad. I actually already divided that I&#x27;m so that&#x27;s equal to 0.425 minus, um, 0.25 c so we can go ahead and simple for this to DC DT is going to be equal to a 0.17 minus C over 40. And so I&#x27;ll go ahead and start a new page here, but we can see that we&#x27;re going to be able to use our separation of variables here to integrate. Um, And so when we separate our variables, we&#x27;re going to end up with one over 10.17 minus C. D. C is equal to won over 40 duty. And so when we integrate both sides of our equation here we get the negative natural log of 0.17 minus seat is equal to one over 40 teeth. And so when we solve for C, you will find that C is equal to 0.17 minus a e to the negative one over 40 teeth. And so to solve for our constant A we&#x27;re going to plug in or initial condition. We know that at times zero see is going to be equal Thio 10 kilograms divided by 1000 liters. So that&#x27;s going to be, um, 0.1 And so when we plug this in, we get the point. No one is equal to 0.17 minus a E to zero power, and we solved for A and get that A&#x27;s equal 2.16 So our final equation is our concentration is equal to 0.17 minus 0.16 a e to the negative or he&#x27;s not a a 0.16 e to the negative one over 40 teeth, Um, and that&#x27;s given to us in kilograms per liter.

Amy Jiang · Answer

Okay, so we know our our out is equal to 60 and our net inflow is equal to 20. Old Firmin. So at time t we have that There&#x27;s 500 plus 20 t. It is less liters of water left in the tank. So our net flow of salt into the tank at Time T is, do you? Why? Over DT is equal to assault rate in which is 80 times 50 and minus our assault rate out, which is 60 times why? Over 500 plus 2080 when I used me for 1000 minus six. Why? Over five for 50 over people&#x27;s duty. Okay, No, be writing this in a form that we do know we get that b y over DT plus six over 50. Plus, Tootie, why is it was 4000. So are a of X is equal to six over 50 puts to tea, and Herbie of X is equal to 4000. Okay, so I&#x27;ll flip to our integrating factories. You got lucky Interval of six times 15 plus two teeth. Oh, over that dj. Well, that gives me E Fri en d plus duty. So yet we can bring this three Make it a power. So we 54 to t you have to me. Okay. No multiplying by 50 postage due to about three. But this we get 50 plus two tea prepared three times y prime plus six times 50. That&#x27;s two t trip are too. Why is equal to for 1000 times 50 for stewed tea to the power of three. Okay, well, that&#x27;s this equal to 50 plus two teams to fire three times. Why crime is he good to 4000 50 plus Tootie Bar three. Integrating both science. We get 50 puts two t drip out of three times. Why is equal to by 100? That beak looks to be about four foot. See, not abiding by 50 plus duty to the power of three on both sides We get why lt is equal to let&#x27;s see 500 times 15 plus to change in the power of one. Let&#x27;s see over 50 plus two two to the power of three. Okay, well, we can mortal play, But if on your end we get y o t is equal to 2 500 plus 1000 t for C over 54 to t to the power of three get, and we&#x27;re also given our initial conditions. That&#x27;s why a zero is equal to 10. That&#x27;s equal to to climb zozo plus one doubts. And oh no T Is there also that zone plus C over 50 to depart three. Gaze ejecting or solving Percy? We get that C is equal to negative. Three points for three one 23 point 75 times 10 to the power of six stood in our initial value problem is Y t 2 500 1st 1000 t yeah, minus. Let&#x27;s say C was 50 plus two tea minus three points or 3123.75 times Center Parcs six over 50 puts to t to power three. Now when t is equal to 20 Fire, what do we have plugging in what we know we have? Why 25 is equal to 2 500 1st 25 times. 1000. That&#x27;s three points or minus three points. No, not 3.3123 27 5 times. 10. The power of six over people&#x27;s too T, which is 25 to power of three. You plug this into your calculator yet. Why 25 is equal to 46876.25 grams. So dividing this way 1000 liters, we get our salt concentration of approximately 46 points. 876 grams. Trey Leader.

Amy Jiang · Answer

Okay, so we assumed that are out. And you go to 1 20 It is for a minute. Mmm Mmm mmm. If our allies 1 20 in our net outflow is 40 it is permanent. So at time, T, we have 500 when it&#x27;s 40 t leaders of water in the tank. Okay, so our question d y over DT is equal to assault rate in which is 80 times 50 minus our salt right out, which is 1 20 times one over 500 minus 40 t. That gives me for now. Then what is 12 wine? Those ideas. Why over him with the money 14. Now, be ready. Is the one that we do know that the wife nobody plus six over 25 when its duty. Right. So we can pull out, uh, two over here, and it&#x27;s you over here that he simplified to six over 25 minutes to t um is equal to 4000 coming by again. All right, so this is a vex or lt this part of beer umpteen. So are Alfa of Tea Alfa. Yeah. All 50 is equal to neither. Integral of six times 25 minutes. Do T minus one That gives me needs anything of three a one of 25 minutes, too. That&#x27;s this. 25 minutes to t to the big three. Okay, Number two playing by that on both sides. We get that 25 minutes. Duty negative. Three. Why? Crime is equal to 4000 25 minutes to t to pounding. To print. I&#x27;m taking the honorable sides. We get that 25 minus two teeth to the power of negative three times. Why is equal to 10 dozen? 25 minutes to G. Shouldn&#x27;t need to. Let&#x27;s see dividing by 25 minutes to teaching in the three Both sides were why is equal to 1000 and one of those and see negative three&#x27;s. That&#x27;s just 25 minutes. Duty See over or three times 25 minutes to t to the factory. Okay, we could expand this a bit. We get 25,000 my next 2000 T plus C 25 minutes to t to fire three. Okay. And based on our initial condition, we know that Why? So you go to 10. That&#x27;s equal to 25,000 Percy times 15 2 out of three that gives a C is equal to negative 1 25 999 So 599 So our initial initial value problem was why she is equal to 25 down when it&#x27;s two dozen G minus 1.599 times 25 minus two T to depart three okay. And were given at time. T is equal to 10. What is our salt concentration? This is why of 10 is equal to 25000 Money 2000 hotel When is 1.599 25 minus 20 two out of three. Can you plug this into calculator yet? Why? Of 10 physical? 24800.8 grams. And then since we went to salt concentration we divide by 1000 three days or 100 meters and we get that this is approximately equal to 48 grams. Very leader. Okay,

Christine Anacker · Answer

Okay, so we&#x27;re onto the work problems of this tractor now. They give you a lot of information in this masculine solved or some different situations. So important things to know the leader. The tapes that were filling with water is about leaders. It is currently filled with 500 years with the salt concentration of 10 grams per liter. Now what&#x27;s happening? ISS that we have a salt concentration that&#x27;s coming in, It says. We&#x27;ve got a salt concentration concentration, a 50 grams per liter that&#x27;s flowing in at 18 years coming. So we&#x27;ve got this concentration that&#x27;s going in and INS mixing instantaneously, which is going to slowly cause concentration in the tank to rise. And that&#x27;s because we are increasing it, have a higher concentration coming, and then we do in the tape. Now. The important part in this question is that our it&#x27;s flowing oat at 18 years per minute. So important to note is that your rate in is equal to your rate, so because I&#x27;m having as much water communities, I am going, we&#x27;ll never actually fill this cake and fire the mountain that take is never going to go higher. Then 500 leaders because as I put in those 80 meters, I&#x27;m taking out eating. There&#x27;s no change in your aura. So what it wants us to do that is by no limiting salt concentration as our time of approaches infinity. So as I&#x27;m going, I&#x27;m starting with this 10 grams per leader and I&#x27;m slowly adding a higher concentration of assault. It is mixing instantaneously and then the same amount is going. So since I&#x27;m not actually changing the volume in the tape, I can actually continue to add water in this 50 grand per liter. Concentration to infinity because it&#x27;s never going to over slow and really, it&#x27;s gonna keep going until I run out of water itself. So really, the limiting factor here is that I keep going to infinity. I am eventually going to approach and maintain that 50 grand concentration. So my limiting factor for myself concentration would have to be if he grabs the meter because I&#x27;m not adding anything that is above that and I&#x27;m not diluting a solution. I am actually causing a I have concentration to go in. We will never be able to surpass that 50 grand&#x27;s leader and because we&#x27;re going to infinity. It&#x27;s never actually going to end up lower than that. So we&#x27;ll always be at the lowest 10 at the highest 50 as we approach. Infinity will never pass that 50. But this is only because in this case, we do have a floor right in and out. That is the same. If we didn&#x27;t, eventually our tank is gonna become or we won&#x27;t actually more water into take it.

Vikash Ranjan · Answer

uh, it well do I need e equals 1 13 minus three. Why are one It was the relational DDO are under. However, it is more one that in my industry my over march minus treaty country 150 we have minus one upon three and then model 1 30 If you see off my as one about three. And then 1 23 You could do to you on 200 minus one of our three and then on one. So it is minus on a country more. 1 30 minus last one for three years, then 1 30 even to dio on 200. No, it is more 1 13 minus three, right? I won t mind. So it stays there. It&#x27;s sequence too. Hello? 1 30 minus two y or 1 90 equal Do minus treaty over 200. No, it would one day my asked by over 1 30 people toe get a power minus tricky. What we have is people&#x27;s too minus C by equal to 1 30 you power minus treaty over 200 minus three equals to 1 30 My 1 30 are minor treaty on it. Are why? Well 1 30 upon tree when my last are minor trading upon did you for? But he had the problem. No, me. How about be well but be we have Why were 1 30.31 minus minus treaty upon 200? Now while 60 is when people to one today upon tree one hour wanted Teoh for the revelation we have? Why 60? Let&#x27;s do 25.71 Find that disease too 25 points to kizzie, which is the answer

Problem

For the following problems, set up and solve the …

Problem 287 Easy Difficulty

For the following problems, set up and solve the differential equations.
A 1000 -liter tank contains pure water and a solution of 0.2 kg saltL is pumped into the tank at a rate of 1 $\mathrm{L} /$ min and is drained at the same rate. Solve for total amount of salt in the tank at time $t$ .

Answer

$u(t)=200\left(1-e^{-\frac{t}{1000}}\right)$

Related Courses

Calculus

Related Topics

Discussion

Top Calculus 2 / BC Educators

Heather Z.

Caleb E.

Samuel H.

Michael J.

Calculus 2 / BC Courses

Integration Review - Intro

Definite vs Indefinite

Recommended Videos

Watch More Solved Questions in Chapter 4

Video Transcript

Gilbert Strang,

Calculus

Heather Z.

Caleb E.

Samuel H.

Michael J.

Integration Review - Intro

Definite vs Indefinite

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$u(t)=200\left(1-e^{-\frac{t}{1000}}\right)$

Calculus

Heather Z.

Caleb E.

Samuel H.

Michael J.

Integration Review - Intro

Definite vs Indefinite

Gilbert Strang,Calculus

Heather Z.

Caleb E.

Samuel H.

Michael J.

Gilbert Strang,

Calculus