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0+71u21H LeeennnSoru 13JOppuanSole dre problem Round your answer; if appropriate:The volume of a rectangular box with square base remains constant at SOOcm?. as the...

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0+71u21H LeeennnSoru 13JOppuanSole dre problem Round your answer; if appropriate:The volume of a rectangular box with square base remains constant at SOOcm?. as the area of the base increases atarate of 140 2/5.Find the rate at which the heleli of the boxis decreasing when each side of the base is 17 cm long: (Do not round your answer)cyscms8 cmscmis10) @uaSoru 14tankata rate of 0.1 cubic metres per minute at the same time the water being pumped into the tank at a constant rate: Thetart Weter Is

0+7 1u21H Leeennn Soru 13 JOppuan Sole dre problem Round your answer; if appropriate: The volume of a rectangular box with square base remains constant at SOOcm?. as the area of the base increases atarate of 140 2/5.Find the rate at which the heleli of the boxis decreasing when each side of the base is 17 cm long: (Do not round your answer) cys cms 8 cms cmis 10) @ua Soru 14 tankata rate of 0.1 cubic metres per minute at the same time the water being pumped into the tank at a constant rate: Thetart Weter Is leaking out of an inverted conical Cnnenhnrennanenelnieamnan4



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The conical watering pail in Figure 17 has a grid of holes. Water flows out through the holes at a rate of $k A \mathrm{m}^{3} / \mathrm{min}$ , where $k$ is a constant and $A$ is the surface area of the part of the cone in contact with the water. This surface area is $A=\pi r \sqrt{h^{2}+r^{2}}$ and the volume is $V=\frac{1}{3} \pi r^{2} h$ Calculate the rate $d h / d t$ at which the water level changes at $h=0.3 \mathrm{m}$ assuming that $k=0.25 \mathrm{m} .$

Spontaneous rate of change of the equation of E f. T equals 100,000 times one minus 1/60 t squared. And to do that, we'll use the equation v prime of a equals limit as X to purchase a of f f experience. F Abe over explain. It's a And for this example, we're finding the general case. So just be prime of t not be proud of any specific number. So we will just use a So to start off, we will have the prime of a equals the limit as X approaches a of 100,000 times one minus 1/60 T Square. And I apologize. This, uh, sex under women should actually be tea because tea is the variable that we're using in this equation. Uh, so that will be that minus 100 1000 times one minus 1/60 a squared. We just replace tea with a all over T minus a Yeah, So if we can take out a 100,000 here and since it's just a constant, we can take it out of the limit altogether. So you're left with one minus 1/60 t squared minus one of, uh, minus 1/60 A squared, divided by T minus a. And when we factor that, um, we will have 100,000 times a limit as T bridges A of one minus 2/60 T plus 1/60 squared T minus one, uh minus to over 60. A plus 1/60 squared a squared. And the sea is also supposed to be squared and all that divided by T minus a. So to simplify, this one and this one cancel up because that's minus one. Still have 100,000 times the limit. S t A. Purchase A of negative 1/30 T plus one over 3600 t squared, minus 1/30. A minus one over 3600. A sward is supposed to be plus 1/38 and all of that divided by T minus a. So now we'll multiply everything by 3. 3600. So sorry. Divide everything with 3600, so we'll have 100,000 divided by 3600 times the limit. As to you. Purchase a of negative 1 20 t plus t squared plus 1, 28 minus a squared, uh, divided by T minus A. So now 100,000 over 3600 comes out to 2. 50/9 has a limit as to you. Purchase A And here we'll take out a negative 120. So we love with T minus a plus T squared minus a squared that of our team minus a. And then we will factor out the last part of that 2. 50/9. I was the limit as C approaches a negative 120 and it's t minus a plus T minus a Times T plus a divided by t minus a. Now all of these terms have a T minus a multiplied by them, so we can cancel that out completely. So your life was 250 over nine times the limit as T approaches a of negative 120 plus t plus a and then we can just plug in T for a So we'll have 250 over nine. Um, sorry. We'll plug in a 40. So, uh, times negative 1 20 ST plus t. And that will equal to 50 over nine. I'm negative on 20 plus two t and that will be our derivative. So that will be equal to V Prime of T. So now it asks us to find a V prime of a bunch of different values. I'm going to go ahead and make a chart so we'll have tea and re prime of T. And the T values are 0, 10, 20 30 40 and 50. And if we just plug in zero for this, we will get negative. 3333.33 If you plug in 10, you get negative. 2777.77 You plug in 20 will get negative 22 to 2.22 If you plug in 30 he'll get negative 1666.67 I'll get negative 40. You'll get negative 1111.11 and finally negative for 50. You'll get negative 55 5.56 and I'll just draw lines to make this a little more clear. Okay, so now it asks us what this means. Generally, it means that the rate of flow is much higher at the beginning and at the end. So the flow rate is the greatest at the beginning and the least at the end. And this is because in the real world, if you have a big container of water and a little hole at the bottom, all the pressure of the water at the beginning will push all the water down. So then the less water there is less pressure. There is, and less hard. The water will come out at the end, so this makes sense in the real world.

Yeah, uh, seems very cool. Quintana when it's very cold Dinner, but training. So you have, uh, sort of like the shape, But this is supposed to be here. Then you have these. They said some lips. Why she have up to these y you love water for some people? He's draining. The water is going out. So you know that, uh, it gives us some reviews. Are that has some height? Why the total volume vehicle too? It would be more for him. The liquid in there is gonna be all right thirds. And this y squared times three. Well, three are Linus. What so well, also very use. This service is these people are here. So for our problem this week are is, uh, 30 meters. We want to find out. Hold fast trees, the level changing, uh, hi equals meters years so that eso or use their release equation. You know that they those over marie quantity, you know that they the water is draining out rate of he's, uh he's quite he's six meters cute. I mean it. So since he's coming out, it's gonna be a negative change into what? So the rate of change there will be negative. So we start with this equation. You know that differentiating that that equation we get differentiating my thirds that can be taken boat Well, these this equation visible toe are why squirt minus what? Cute. He's piece here about Bert. So, uh, appreciating that well, they are bigger is a constant. So you get the bones we are and two too high. You want tea? You are the tea Then appreciating that part minus three. What? You know why? So that people security the three Do you want anything? This will be ableto bye times Are that was why No Also pencils What? Forget the squirt. Why square quit my nose. Why quick? That won't buy by you. Why? Um So that they you are The team will be ableto well. You have your your changed. You brought him here. Execute the drinking water. My six meters Two You mean it? That divided by it's quantity. Bye. Thumbs are so two comes Why so terms? Why? So how's qualities of meter square minus square So that the council you get meters per minute Get this year old bean um, vehicle too. So 13 2 of them's a for two years. Terms 16 minus you squeeze 64. So you know, I get minus six over what we get fucked Her one made No will be easier. This quantity here come very close Their tears. I'm still minus eight so I won't 14 times two. That's 26. 26 knows people too. Six more rooms. A team. It's gonna be team. I was eight. So so is your vehicle too? Yeah, Well, why 18 a. Uh meters, I mean it. Eso Well, it is able to feel I was six. So that's six full council six. You can write that us mine does one over so three times a beast thing for bye. Looking comes before years. So we have. This is the rate of change of Did Did that eso Well, we have these play angle They are is related You again our serial shape. You have some radios. So my radius this little are is related to what work? Why is it these things? He's a ll these times will be We are so that you have the triangle are minus why we're here. We'll be besides games are I'm being well, It could be Excite our Yeah, Set up for the strangle. Well, our square movie photo are Wait this way. That s good. Plus are square beetle are square So So I'll give you square. Dad, it's my bm square witness through our Why, yes. Why? Word that she brought home square game You have square So from this equation can counsel square you can move not right there. He's there was obtaining that two. Are you got it? That's why minus why square should be able to well are square So from this relation You know that um knew our movie photo we're related to Why for big are being equal toe 13 in our problem 30 meters This would be ableto good times 13 so six vehicle to queen six Why minus why square well then square with that mark is here that the depth why cannot be any bigger And your race No Make sure that he's I wanted to hear inside This crude is a positive number there Well also from here you get that with the friendship, his relation You know that you are two terms who are from appreciating this relation should be able to Well there. Two are to be our Why team minds? Why? Minus y you what? You too. So these the r Did you if you cancel the twos, he already thinks I'm ableto are minus y. Why the team over little or so indicates that we have. Well, these radius is 13 from the depth here. So these remaining that difference 15 see? But it's gonna be me five on, uh well, we know that 13 squared should be equal to Live Square. Who was that you to get, huh? Then that to get weather should be there. Right? Change in there a little Are the radius respect? Oh, he's with change. I said I know. Well, he's, uh, little are have ableto 13 square weariness. I'm square. So Art square's gonna vehicle too. 15 years. Quarries. One tonight, Like squares. Right. So these numbers one for just go two square. So they are. Is he going to grow and we have or rates there? It'll change the little off the radius. Photo live over. Well, terms you want it. But she's, uh, minus work over. So comes money won over. Bye. Who's 24 readers? No. So these quantity, it's their loss. How fast is that? Brady's big person

So for the problem if we have a cylindrical tank that holds 100,000 gallons of water, we have volume given to us here as a function of time. So v. is equal to 100,000 um times one minus t over 60. Yeah squared, It's gonna be from Europe to 60. So we want to find the rate at which the water is flowing out of the tank um as a function of time. So that's gonna be the prime of T. Which is going to look like um The property is going to be 200,000 times one minus key over 60. And then that whole thing is going to be divided. I am -60. You can tell it's the rate of change.

So we're told that there's a hemispherical reservoir, so it looks like this is half circle these ears. The hemisphere's see your It says that, um, the volume is given by V. Is he gonna pi over three? Why? Squared times three R minus. Why? Anywhere. If you're sort of thinking of this as the hop of the hemisphere in this right here you go down or sent to hide the water. I guess, um, this is the diagram is actually in your textbook. We get 13. Here we have. Why here? Which is the height of the water level. And then this is our here. Okay. Yeah. House is something that the water it's flowing at a re out of it. So every I've changed. A volume is minus six years, que per minute, and it says the radius R is 13 k, so we can put that in there. This is the answer. The following question. So what were the one little changing when the water is eight meters deep? So it tells us why is eight meters that want d y by d? T for acting for anyway? So we're gonna do is gonna say the volume is by over three y spared on through and then three error is actually 39 13 buying twice. So what I'm gonna end up with is the zeal of pilots to be on who's 39 wives fared minus. If that's all you and then I'm gonna take the derivative of that. No TV by t is going in a pi over three multiplied by the dream of this because you're gonna bring the two gallons or two times 39 it's 78. Why do you? Why, by g getting minus three wide clearing the White Plains like you though I'm gonna start claiming my numbers in, um D V by d t. First of all, if only was minus six equals, then I know the fire or three is a constant, though it's just going to be there. And I know that the value for why they only with heat Okay, let's just be why over deny do I equals TV, So I'm gonna go eat for why so eight times 78. Um, I'm seeing why d and then minus three times wide square, which is eat very wide body. All right, that's that feels in my number. So then I gotta figure out that equals by minus six is equal to pi over three on. Do you know why? My ET is actually a common factor and a video eight times 78 first with more 3 64 And that is 192. Okay, that minus six is equal high over three multiplied by text 24 minus 192 which is 432. And you don't necessarily have to go through every one of the stepped. But I am then 432 divided by three is 144. Look at minus 144 by the wind by key to that means if I saw her d y by d t is equal to minus six over 144 pi and that reduces to minus three her 72 pi 80 different reminders that we will go to first. I mean, actually that we could buy The three is 24th you experience with minus 1/24 pi and the unit for why is meters meter for and then the on your this minute so that's d Y by the year before pie. Now, the question is at that point, go back here. This is our here. All right. This is gonna be 13. Mine it. Why? Okay, so we had 13 minus y squared, plus R e swears he was 13. Good. Uh oh. If I take the dreaming that get you time. 13 miners. Why minus d y by D. That's the derivative of 30. Minus y was to earn The are biting e is good Equals zero. Now we know why, right? We know why we're good there. We know d y by d. T. We're looking for a year by year, but are when I go back up here for a second. If I take 30 minus eight squared lost ours. Where is he? Was 13. Spared that some of that five players, but are knitwear with three people and hopefully recognized those numbers. If you don't, you can solve it. Um, which would solve like this because 12. So now he's a plumber. Numbers field 13 minus times minus B y by even with it. 1/24 pi. Okay. And then plus two times 12 times the air by DT, and that equals zero. So this is gonna give me, um I just do it here sometimes. Funny. Which it's 10. Or you can do this. You can make this. Make it. Go and buy. It's five over 12 pie. Okay? What? 24 the are biting is equal. This euro. So then if I continue that minus 5/12 guys that equal you if you jump back here minus 24 year by D and so I'm going to get minus five over whenever 12 before is okay if you look sorry. Yes, I am. No. 12 times 24 is going to be 288. We're in one in 88 by and the Indians are actively be here is going to file this pretty pie meter per minute. That's the our body team. Okay, So actually, part be asked you just to find the radius, which is this part. And then part C asked you to do this part. So there's your answer


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