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25 gallon tank initially contains 16 gallons of [resl: water. salty solution con- taining pound of salt pCr gallon enters the tank at rate of 5 gallons per Ininute:...

Question

25 gallon tank initially contains 16 gallons of [resl: water. salty solution con- taining pound of salt pCr gallon enters the tank at rate of 5 gallons per Ininute: The well-mixed solution in the tank is then pumped out of the tank at rate of gallons per minute. Find the amount of salt crreC[ I0 (ILO' decimal place) the tank a tlie instant the tank overflows_

25 gallon tank initially contains 16 gallons of [resl: water. salty solution con- taining pound of salt pCr gallon enters the tank at rate of 5 gallons per Ininute: The well-mixed solution in the tank is then pumped out of the tank at rate of gallons per minute. Find the amount of salt crreC[ I0 (ILO' decimal place) the tank a tlie instant the tank overflows_



Answers

At time $t=0,$ a tank contains 25 oz of salt dissolved in 50 gal of water. Then brine containing 4 oz of salt per gallon of brine is allowed to enter the tank at a rate of 2 gal $/ \mathrm{min}$ and the mixed solution is drained from the tank at the same rate. (a) How much salt is in the tank at an arbitrary time $t ?$ (b) How much salt is in the tank after $25 \mathrm{min} ?$

Hey, it's Claire. So when you right here. So for per inning, we know that Brian containing two go to Pums for down of Soul is running on a team at five gallons per minute. So we're going to do to kinds five only end up getting £10 per minute. Your part B We're finding the volume. So we get free of tea, be equal to 100 plus guys t minus 40. So this is equal to 100 less be gallons. Now, for part C, we have the volume, Ms. V of t is equal to 100 plus t. So we have the amount of salt in the tank and we're going to find the concentration at any time. You know that? That's why over 100 plus t, the lie of tea over the of tea is equal to why over 100 plus t homes break Alan tons for which is equal to four. Why over 100 plus t. Then we get Do you? Why? Over d t. This is a plus for over 100 Blust t to the wind power people to turn. So when we do e to the integral of four over 100 plus e t t we 100 plus t to the fourth power and we get 100 plus t to the fourth power you buy over DT plus 100 plus t to the fourth power before over 100 plus t to the white power equal to turn times 100 plus teen to the fourth power. Then we have 100 was t to the fourth power Do you lie? Plus 4 100 lefty. Cute. Why do you thi which is equal to turn 100 plus four 100 plus t to the fourth power d t Then we make sure that integrate both sides and we end up getting a why Value of two. I'm 100 plus t bless. See over 100 post T to the fourth power. We know that the that's the interval iss there. Oh, comma 50. That's the X for you and the y value. So we get 50 is equal to two times 100 plus zero plus c over 100 cereal. The fourth power we're gonna see value with negative 1 50 I'm 104th power. No, we just replace it in so yet war is equal to two turns. 100 plus he minus 1 50 over 100. Plus he over 100 to the fourth power. We're gonna plug in 25. We get about 188 0.56 for parts her e pounds. So we're going to calculate the volume after 25 minutes. To get rear t is equal to 100 less. So you get one of your 25 gallons. So we do 100 85.56 over 1 25 is equal to 1.5 gallon crowns per gallon.

Longer. Eighteen. We have a tank holds one hundred gone off water that contains twenty pounds off different thought. A brain solution is flowing into the tank rd right off two gallon per minute while the solution flows. Our told SID hand at same rate. So bring solution. Interim C Tang has sowed concentration to pound Fergana. OK, find an expression for their monarch sewed in the tank at any time. So we can assume Teo. Why? To live the amount. Wolf out t that time in a minute. A minute? Ah, so d y over dt is a chance. He's a red off change of the Armando's out, which means youcause Assad come in to solution subtracted the I thought close out for the solution. So how many's outcome in each minute we have that? Ah, a brain solutions flowing into the tank at the rate of two gallon permitted and that bring solution entering the tang has assault controversial to pound per gallon. So too gallant, too pantos Opera Gannon, We got full pounds of salt every minute. Come inside off the attack. So how many's out? Flows out? Ah, so we have the amount ofthe out. Why in the tank a time tea and two ten has one hundred gallon water. So that's the density. And you flows out Two pounds to gunnery clothes are the same, right? So to going back to that's having south close, are you? Two minutes. So it's going to be four miners. Why over fifty? And we can solve this equation by separate ever bust this one over. Full minus Wilder fifty t law people was tt Hey to grow, we have We've come only about fifty four fifty two hundred marks. Loy Okay, it's gonna be a lark. Two hundred minus y fifty. Negative to that one over is what father to fifty thatyou want. Yes, because t classy. So let's lock two hundred minus y if wass conectiv t over fifty plus some constancy. Ah, two hundred modest wa It was It was inactive t over fifty. Yeah, but classy. So we got Why? Because two hundred minus c modify e to the t over fifty and all her initial body. Twenty pounds. When do you go? Zero So we can plug in twenty twenty pound. It was two hundred miles T was there of minus e, so c was one Katie. So our function final Tompkins Why was two hundred minus one eighty e to the off with his not direct negative t over fifty. There's our final function. And he's how much salt is present after one hour. So after one hour, mystique was sixty sixteen minutes. Why was two hundred minus one eighty multiply you today? Elective? Sixty over fifty? Yes, His calculations done. Sure. Do it by five. Yeah. One forty five. It was one forty five point seven eight five pumps off south. After one hour a time increases what happens to decide concentration. So as time increases means the time goes to infinity. Why was two hundred minus one? Eighty e t goes to infinity? That goes from factory infinity. What goes through that you two the elective infinity, Remember? Thanks. You do selective something. Something like that. Function like that. This Well it does today aRer so why goes to two hundred? Ask t goes to return. It means the concentration goes to show how one hundred pound off Galan Mondragon off watering that hand. Close the time because it right off. Ah, the rate flow out of the tank of the same rate as the what you're following inside the attack. So we always got along begun on water. So your concentration he's too town per gotta two pounds per gallon when time increases goes to yoga? Because, yeah. Ah, to bring has a concentration of two pounds began. And if you keep, put those solution inside and you are too gone off. Water sucks. Illusion remain. It is going to be getting closer and closer to that concentration. All right, so which made perfect sense.

We're told that we have a saltwater concentration of 0.1 lbs of salt Per gallon flows into a large tank that initially contains 50 gallons of water. And so if the flow rate of the salt water into the tank is five gallons per minute, find the volume of water and they're mountain um Of salt in the tank after t seconds, 40 minutes. So the volume, the total volume here is simply the initial volume plus this added stuff. So it's again it's five five gallons per minute. So we have five t. So this is the total volume After a certain number of minutes. So we start with 50 and then we're just adding some salt water. And now the amount of salt um is simply, uhh so it's it's it's um £0.1 her gallon of Put White, one lb of salt per gallon of liquid that close in. And so we have five gallons per minute. So we have 0.5, so 1/10 of this, 0.5 times t is this is the amount of salt. So again it's it's five gallons per minute and 1/10 of a pound per gallon. So we have 05 gallons or no pounds per minute of salt going in. So that's a salt content. Salt amount of salt is obviously increasing because we're putting salt water in. So then they said find a formula for the concentration where the concentration was just the amount of salt divided by the total volume. And after a little bit of algebra when you find that that is t all over 100 plus 10 T. Then they asked excessive variation of CST coast infinity. Well as he goes to infinity, this thing this term gets large compared to that. So we can neglect that and then the tears drop out and we see that the concentration goes to um 1/10 per gallon, which is what basically the concentration of what we're putting in there. Because as he goes infinity as we keep filling, filling, basically this initial amount here becomes very negligible. So the concentration approaches the concentration of whatever it is we're putting in, which is 1.1 lbs per gallon. Yeah.

Okay, so for this problem here asked the first question says, How much salt is in the tank? Arbitrary Time t. So we're gonna be looking for D Y over DT, which is going to be our salt level. So why being are the amount of salt and anti being time? So the first part tells us a tank initially containing 200 gallons of pure water. So that tells us when T is zero. The amount of salt in the tank is zero because it's pure water, then it t equals zero. Brien, containing £5 of salt per gallon of Brian is allowed to enter the tank at a rate of 20 gallons per minute. So we have entering. We have £5 um, of salt per gallon at a rate of 20 gallons per minute. So this is gonna be important because we want this to equal, um, in minus out. So then the last little piece of information just tells us that it is The make solution is drained from the tank at the same rate, so that just tells us that the rate for each of these is going to be the same. So the rate going in is going to be £5 per gallon times a rate of 20 gallons per minute. So it's £5 per minute minus the amount leaving, which is unknown over the 200 gallons Hank times a rate of 20. So now what this tells me is that d y over DT is going to equal 100 minus 20 divided by 200 which is gonna be 1/10. Why, So now what I'm gonna do is I want to set this up in the linear format. So d y over d t plus 1/10 y equals 100. And then what I want to do is I want to solve this. So I'm gonna find my, um, my integrating factor. So it's gonna be e to the integral of 1/10 tee. I'm sorry, DT Nazi. So this is just gonna be e to the t over 10. So instead of actually adding this to each one of these, I'm just gonna going in straight into this, so it's gonna be d over DT, and it's going to be e to the t over 10. Why equals 100 and I must set it up is an integral So d um eats of the tent t over 10. Why equals the integral of 100 DT. Now, what I want to do is I want to go ahead since thes canceled. Gonna bring down E to the t over 10. Why? And then, of course, this is gonna be 100 t looks. I never got my e to the t over 10. So it's gonna be e to the t over 10. And then I have times the reciprocal of the 1/10 tee and then plus a constant. So then I'm gonna have y equals 1000 um, plus C e to the negative t over 10. And since I have an initial condition of why of zero equals zero, then I have zero equals 1000 plus see Edith zero, which of course, is just one. So it looks like C's gonna equal minus 1000. Someone I have. Why equals 1000 minus 1000 e to the negative. T over 10 is part a. So this is what the amount of assault that's in the mixture at an arbitrary t. So then, for be what we want to dio is. We want to find out what's how much salt is gonna be in there after 30 minutes. So it's gonna be why, if 30 equals 1000 minus 1000 each of the negative, 30/10 so e to the negative three, which is approximately going to be £950.21.


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