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Waterleaking Onio floot creates circular pool with = inciejit rate of 3 square inches per minute. How fast the radlus - the pool increasing when the rudius 10 inch...

Question

Waterleaking Onio floot creates circular pool with = inciejit rate of 3 square inches per minute. How fast the radlus - the pool increasing when the rudius 10 inches? n A= LonTwo airplanes are flying in the air at the same height; airplane flying east 250mi/h and airplane is flying north at 300mi/h. If they are both heading the same dindon_ located 30 miles east of airplane and 40 miles north of airplane at what rate the distance between the airplanes changing?26

Water leaking Onio floot creates circular pool with = inciejit rate of 3 square inches per minute. How fast the radlus - the pool increasing when the rudius 10 inches? n A= Lon Two airplanes are flying in the air at the same height; airplane flying east 250mi/h and airplane is flying north at 300mi/h. If they are both heading the same dindon_ located 30 miles east of airplane and 40 miles north of airplane at what rate the distance between the airplanes changing? 26



Answers

Water is leaking out the bottom of a hemispherical tank of radius 8 feet at a rate of 2 cubic feet per hour. The tank was full at a certain time. How fast is the water level changing when its height $h$ is 3 feet? Note : The volume of a segment of height $h$ in a hemisphere of radius $r$ is $\pi h^{2}[r-(h / 3)]$. (See Figure 12.)

Okay, We know we have V equals pi H squared over three times three are minus h. Taking the derivative. We get Devi over. DT equals pi age. Times two are minus h d h over. Did t we know that one interest per second is the same thing as five feet per minute. Right? We've got two different units. Inches and feet swift be consistent. So playing him what we know two times turn minds too. Times five is equivalent to 180 hi feet cubed per minutes. It's Stevie, which is volume. Yeah. So one thing you wanna note for this is the unit conversions. 12 inches is one foot 60 seconds is one minute. So you have to do the unit. Conversions are also end up with the wrong number, even if your calculations are correct.

So we have these two planes one going 250 miles per hour towards the airport and another going 300 miles per hour towards the airport where a is going east and bees going north. And we want to find the rate of change of the distance between these two planes when a is 30 miles away and B is 30 b s 40 miles away from the airport. Now what we just have drawn here? Well, it looks like we have a right triangle. So we might want to find some way to shoehorn a triangle property into here. So if I were to call this side of the triangle X this side of the triangle, why and this side of the triangle See, we could use Path a gris to relate all three sides to each other. We have X squared plus y squared is equal to Z script and what we're looking for in terms of X, y and Z is Deasy by D. T. Cause we want to know the rate of change of that red line or the rate of change of seat. Well, if we were to take the derivative of X squared plus y squared is equal disease Weird. We will end up getting dizzy by DT So let's go ahead and do that really quickly So D ready t So all these will look similarly so the derivative of X squared would be to x and then we would have to multiply this by D expert e. T. Because of chain rule were implicit differentiation because X depends on t And then we get the same thing for why so too I times d y bi tt And then this is going to be equal to two z times easy by TT. And now we could divide just by two to get all those accounts alone. All right, so now, before we do anything else, let's figure out what x d exp i d t and why and d y by d. T should be. And we should probably also figure out what Z is as well. So in this case, Z is supposed to be the distance between A and B. So then that means accent. Why would just be the distance between the airport and each of the Plains? So X is going to end up being 30 and then well, the rate of change of this should end up being DX by d. T here. But since the distances decreasing, this should be negative. 250 that we use and then likewise for playing be This will be our why. But then d y by d. T. This here should be negative as well. Since the distance between the plane and the airport is decreasing and will define Z well, we would just need to use Pythagoras again. So go back to X squared. Plus y squared is equal to Z squared and we can plug in that X squared or excess 30. So scoring that would give us 900 and then why was 40 so scoring that would give us 1600. This is even twosies square and well 900 plus 16 hundreds 2500 And then we would want to square root each side so we get Z is equal to plus or minus the square root of 2500. But it doesn't make sense for us to talk about the negative route or Z in this instance, because this is a length between A and B, and it doesn't make sense to talk about a negative distance between two things in this instance. So we have that and then we can actually simplify this to just 50. So we know all of our values for everything outside of Easy buddy t. So let's go ahead and plug everything in. So we said X is 30 DX by DT would be negative 250 And then why is 40 and then d y by DT is negative 300 and we had Z is equal to 50 and then he would have Deasy by d t over here. So the first thing I'm gonna do is just divide everything by 50 because I don't help simplify the numbers a little bit. So doing that would give what we have 30 times Negative five waas 40 times Negative six and this would be equal to easy by DT. I don't want the equal sign here and now we can go ahead and multiply those together so 30 times negative five would be 1 50 and then we add that to 40 times negative six, which should be negative to 40 and adding those together we would get negative 3 90 so minus 3 90 this should be in miles per hour since our other two units for velocity, or, I guess, in this instance speed or in miles per hour. So the distance between these two planes is decreasing at a rate of 390 miles per hour.

So for this problem, we have to swimming pools that are being filled at the exact same rate. All right, so we have a small pool and we have a large pool. Okay, What we're told is the radius of the small pool is five meters and the radius of the large poll is eight meters. They were also given the information that the height of the small pool is rising at a rate of 1/2 meters per minute. We're trying to find how fast the height of the larger pool is changing. Okay, well, we're gonna have to do a couple of problems here. We can't because we can't go directly from the small pool to the large pool. It's what we need to dio is we need to find out how fast the small pool is being filled because we need to know that to determine how fast the height of the large pool is changing. So we're gonna have the volume formula of a cylinder, and it is pi r squared. H right. So we do have one constant. Our radius is constant. So I'm gonna go ahead and plug that in for our our is 55 squared is 25. So I'm gonna have V is equal to 25 8 Sorry. 25 pi age. There we go. Okay, So we're gonna take the derivative of both sides with respect to T. So DVD T is equal to 25 pi d h d t. And then I'm gonna substitute in 1/2 here. All right? So DVD t is equal to 25 pi over two, and that's cubic meters per minute. Okay, So now I can come over here to the large pool side, and I'm gonna use the same formula. V is equal to pi r squared age Once again we know pie There. Sorry. Are is eight so eight square to 64. So V is equal to 64 pi h eso once again derivative both sides We have DVD t is equal to 64 pi d h d t. This time we know DVD T, which is 25 pi over two and we're trying to find the h d. T. So I'm just gonna divide out 64 pi and I end up with the H d. T is equal to 25 over 120 aids meters. Her minute

All right, we've got a big pool, it's 20 ft by 40 ft by six ft. Uh And then water is leaking. So water is leaking by this formula W. F. T. Is 35,000 Uh -2 T. Square. That's the amount of water in the pool. Um Where t. Is the number of times since the pools last field? So at what rate is the water leaking when T. Equals two. So this is amount of water. So if we take a derivative we get the rate of change of the amount of water. Um So derivative of 35,000 0. And then minus, we use the power roll, bring it to down in front to get a. Four. We have a T. Here. Um And it's negative so that's good because it's leaking that's consistent and we need to find w prime of two at 2 hours. So that's um negative eight. So it's and then we were in gallons and hours. So the water is leaking At a rate of eight gallons per hour When he is two hours.


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