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A tank initially has 200 gallons of a solution that contains 25 Ib. of dissolved salt: A brine solution with a concentration of 2lb of salt/gallon is admitted into ...

Question

A tank initially has 200 gallons of a solution that contains 25 Ib. of dissolved salt: A brine solution with a concentration of 2lb of salt/gallon is admitted into the tank at a rate of 4 gallons per minute. The well-stirred solution is drained at the same rate How long will it take for the tank to have 100 Ib. of dissolved salt? Round your answer to the nearest minute.

A tank initially has 200 gallons of a solution that contains 25 Ib. of dissolved salt: A brine solution with a concentration of 2lb of salt/gallon is admitted into the tank at a rate of 4 gallons per minute. The well-stirred solution is drained at the same rate How long will it take for the tank to have 100 Ib. of dissolved salt? Round your answer to the nearest minute.



Answers

Salt Concentration A tank holds 100 gal of water that contains 20 Ib of dissolved salt. A brine (salt) solution is flowing
into the tank at the rate of 2 gal per minute while the solution flows out of the tank at the same rate. The brine solution entering the tank has a salt concentration of 2 lb per gal.

(a) Find an expression for the amount of salt in the tank at anytime.
(b) How much salt is present after 1 hour?
(c) As time increases, what happens to the salt concentration?

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.

Okay. Okay. Trinket is Herman. We find it from the rear, which to sell. Interesting. Which is shown to be the real reading is art is equal Dude town carry by Heard me, which is then equal to 10. How her buying If this one's equal Teoh volume is equal to equals 100 Ah, 100 gallons of brine waas. Whatever the differences, which is five Mainers. I've t is 14 team downs There are now you got a two term the part c we're determining that were determine the rate, which I think so That's why he's the amount you represent use why teaching represent the Memphis soul? Uh, representing the represents the amount of salt that leaves it hanging divided by following you just equal to, uh for why we tripped Installing assault about 500 plus t in which white t b people to you, or why you need multiple bad things. They know that it leaves out the read. So that's how much salt will be leaving here now. The initial value problem would simply be Yeah, so that won't be 1/3 party. It will be necessary for beginning with this from now on yourself for? For younger gets them from D do, do you? I i ke is which is important too. 10 minus for why I 100. Plus she This is the difference in the first round differential equation. Allis community arranged to be looking like t I for why 100 Plus you really equaled why his equaled two times 100 e e waas You voted by 100 to the for? Not for her. You can actually first. Why? Why? But 25 which is supposed to be able to 180 six. And then you told a 25 minute which is a hiring so the concentrate them would shrink. We trying the concentration would be. Why 25? If I write the meat by, we should teach one approximately 1.5 Oh, wait £5 per gallon.

Okay, So for this problem, we're told that at time equaling zero a tank contains 25 ounces of salt dissolved in 50 gallons of water. So for our why of zero, we have 25 pounds. And then, of course, we can kind of say that this isn't 50 gallons of H 20 So then it says the then Brian containing four ounces of salt per gallon is allowed to enter the tank at a rate of two gallons per minutes. So we have entering four ounces per gallon with a rate of two gallons per minute. Okay, so what I want to do is I want to Duthie rate of change. So it's gonna be d y over DT where why is the mount? And obviously T is time. And so it's going to be the rate in, ah minus the rate out or rate out minus rate in depending on the situation. So for this one, what I want to dio because I want to say the rate in So I'm starting off its force. What's going in in four Alice is per gallon at two gallons per minute and in minus my the amounts over 50 cause it's a 50 gallon tank times the rate So d y over DT is going to equal eight minus Why over 25. So now what? I want to dio because I want to set this up for linear set up solving using a linear solution. So what I want to dio is I want to add 1/25 y equals eight. Then what I want to do is I want to find my integrating factor. So I'm gonna have e to the integral of 1/25 DT. So that's going to equal e to the 1/25 t. So then what I know is that my d over d d t is going to be the integrating factor times the variable which, um, is going to be Why equals eight DT. Then I have to add put my integrating factor. So now if I'm integrating each of these, I know that this is going to cancel. Its gonna leave e to the 1/25 tea. Why? And so now I'm gonna have my eight and then eats of the 1 2050 and then I have to do the reciprocal of what the constant is in front of the tea, which is gonna be 25 and then plus my constant I'm gonna divide by E to the 1 2015 So we'll be left with y equals 200 plus c e to the negative one 25th t. So, since I have an initial condition, which we said is why zero equals 25 then I can go through and plug this in to find out my constant. So, you know, eat of the zero is one, so it looks like, see is going to equal negative 1 75. So my equation is going to be why equals 200 minus 1 75 e to the negative t over 25. So we were asked to first, um, find the equation and find it for an arbitrary T, then were asked to find what it is after 25 minutes. So then what I want to dio is I want to find why of 25 so it's gonna equal 200 minus 1 75 e to the negative 25/25. So putting this into a calculator, it's gonna be approximately 100 35.62 and then we're talking about ounces

So we'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'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're pumping it out at a rate of 25 liters per minute. But the concentration of the assault that's being pumped out is going to be equal to the amount of salt in the tank. I'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'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'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'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've already calculated it. Um And so essentially, we're just going to plug in 1000 for V. So when we do that, um, we get DC DT is equal to I'm in this step, actually already divided by 1000. I'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'm so that'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'll go ahead and start a new page here, but we can see that we're going to be able to use our separation of variables here to integrate. Um, And so when we separate our variables, we'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'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'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'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's not a a 0.16 e to the negative one over 40 teeth, Um, and that's given to us in kilograms per liter.


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