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Question BWater is pumped out of a holding tank at a rate of Se 0.12t literslminute, where t is in minutes since the pump is started If the holding tank contains ...

Question

Question BWater is pumped out of a holding tank at a rate of Se 0.12t literslminute, where t is in minutes since the pump is started If the holding tank contains 1000 liters of water when the pump is started; hOw much water does it hold one hour later?Question €Use the information given in Question B to calculate the pumped from the holding average rale at which water is tank over the first 30 minules.

Question B Water is pumped out of a holding tank at a rate of Se 0.12t literslminute, where t is in minutes since the pump is started If the holding tank contains 1000 liters of water when the pump is started; hOw much water does it hold one hour later? Question € Use the information given in Question B to calculate the pumped from the holding average rale at which water is tank over the first 30 minules.



Answers

Water is pumped out of a holding tank at a rate of $5-5 e^{-0.12 t}$ liters/minute, where $t$ is in minutes since the pump is started. If the holding tank contains 1000 liters of water when the pump is started, how much water does it hold one hour later?

For this problem. We want to use the results of exercise 22 to find the average rate at which the water runs out in the different intervals and from exercise to We found that the difference quotient is negative 8000 plus to t plus h and that is going to you how we find her average rate of change. And for part, a R interval is from 10 to 10.1 minute. So our tea is 10 in our H is 0.1. So if we plug those in, we get negative 8000 plus two times 10 close 0.1 in that equals knighted 7979.9. For Part B R. Interval is from 10 to 10.1 So again, our each is the value that changes. So we get negative 8000 plus two times 10 plus 0.1 and that equals negative 7979 quite 99 And so, if we were going to estimate the rate at which the water runs out at exactly 10 minutes, it would be negative 8000 plus two times 10 which is about 7980

So this problem We're getting crazy. How much would you pay to tank out of state? And the first part is me to find how fast the water running out in 10 minutes. So you're squeezing your teeth and to find that's great. It come here. I think. First we take the derivative of beauty to the first step. You need to expand cute. So if you multiply, um, expand. You get cute. Tea cools 180,000 minus 12,000. Teen plus 200 square. No, you need to take the derivative a few teeth. And we would get Meghan years. 12,000. What? Or 100? No to find. Um, the rate that is expanding at actimates you simply plug t about 10 in for two, so get made it in 1000 bus for 100 times 10. And if you multiply and at that we would get negative 8000 gallons a minute. Perfect. So the second part this problem asks us two find the average rate at which the water folk like you in the 1st 10 minutes. So the average is the salvaging two points and in this case, are 2.2 B T equals 10 and t equals zero. So we're basically finding object change form hue of 10 minus Q. Zero over 10 minus through. So if you're playing you of 10 you get 200 but my square minus 200 times square. You put that over. This is really And if we, um, simplified die that equals 200 times 400 minus 200. Tom's 900. And then we get that all over 10. And if you finally simple fight that you get a final answer. Negative. 10,000 Joe is permits. Have yourself a problem.

We have a cistern that is getting filled. The sister and can hold up to 2000 leaders and it starts empty were also given the rate of change the Q prime a. T that the the water is coming in. First, we want to know how much water has has flowed in in an hour or 60 minutes. Well, if we integrate the rate of change, that will give us the amount that's flowed in. So I'm gonna have Q of 60 equals CNN girlfriend 0 to 60 of my Q prime. Don't separate this into two columns. The integral of that would be, well, that's three to the 1/2 power. And if you add one to the power, that's going to be the three halves power. And then I'm going to divide by three halves or multiply by 2/3 and three times 2/3 is two evaluated from 0 to 60. If I fill in 60 I will get about 929 and 1/2 liters, and I would subtract What I what I get when I fill in zero. But that's zero now. I did start empty, so if it went up by 929.5 leaders, that is, it's new amount of water in it. Next, we want to figure a formula to represent the amount of water at any time. So we know that Q. Of T is the integral of Q prime of tea. So we're going to integrate three times the square did to your team to the 1/2. And just like we did in the previous problem, when we integrate that we're going to end up with to t to the three has power. And that would still be plus a constant now to figure the constant we know that to be getting when the time is zero. There is zero leaders in that, and if we happen to fill in zero for tea here, we actually end up with zero plus see on the left. So if we start with an amount of zero, that indicates that C is going to be zero. So the formula for the water any time is too t to the three s power family. We want to know when this is full, or that is when it has 2000 leaders in it. So what's fine. When Q of t is equal to that, I have 2000 equals. Two t 23 haps fine. Divide by two than tea to the three halves. Power equals 1000. And then, to get rid of the to third power 33 house power, I'll take both sides to the 2/3. So tea is 1000 to the 2/3 power, and that ends up being 100. I remember that our time was in minutes, so it's going to take 100 minutes to fill.

So there's problem. We're being told that we have the tank here like it's losing water and well being that's being moved by this function here, right? Cute. He is just currently the mount that woods in the tank starting at time T equals zero. So for the first part of what we're being asked, what is the rate of flow? Um, out of the tank? But teak was Tenerelli. And as I've labeled quite nicely here, they're basically asking for Dick You, Right, So, to find a Q, you just take the directive and respect T. So we do that. You get fined the Sequels 200 times, two times 30. What is T terms native according to her cause of the material. And this becomes 400 times T when a certain and then we can plug in t equals 10 because we're house importance that Tim, you got 400 Most wanted by the year 20. So negative. Basically, we're losing 1000 so 8000 close out. Right, So you take the dirt that officer value causes office. What's the average rate of flow out right? We're losing 1000. So 1000 fully now, so this would be 80,000 gallons per minute. What time? Tea was 10. And then we're being asked. So he wrote this, uh, whole thing. What is the average rate of flow out? My knife? As I have also nicely labeled here. It's basically the Austin else. You know, the changing Cuba is a similar to Robert Frosty for younger Children. Take essentially Thea. The final amount minus the initial mountain. You know, Collins in the tank over the, um, final time in the initial time. So changing number of gallons it over changing time. Right. So to do this so you can just essentially click in our values that teak was 10 right alongside. Given, given this equation here, so cheap was 10. You have that Finals 10 and financial sense of disability. However, um, you plug in no t equals 10 for a Q t. We find that at T zero. It's just gonna be 200 multiplied by 900 because already squared is 900 and this is essentially 18,000 so I'm a righty. Chan k over here. Um, this is gonna be a Q and Q initially. So that's initial. Um, however, que final began Pollyanna. 10 for this value right here. Right and weak. I thought this was gonna turn into 200 multiplied by 20 squared. Right. So let's 400 and this basically turns into Okay, I I meant 180 were here. Right? So now that you know initial and no final we're here, you can find that 80 k minus 180 k gives you 100. Okay? Right. And then repeated by 10. You find that this was 10 kids. So you see, young Oh, my country myself 10-K right from then shows us that the average rate of flows 10 tape for a minute. The reason why I se Jin Qena minus 10 days because they were again with more flows opposite values. So, yes, I took the absolute bottom here of what it would have been which were negative. Yes,


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