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Solve the problems in related rates.An engine cylinder $15.0 \mathrm{cm}$ deep is being bored such that the radius increases by $0.100 \mathrm{mm} / \mathrm{min} .$ How fast is the volume $V$ of the cylinder changing when the diameter is $9.50 \mathrm{cm} ?$

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 4

Related Rates

Derivatives

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question tells me that we have a conical reservoir that is holding water. So we're gonna assume because it's holding water, that the code is like this. And so there's getting water in the comb. All right, this is my water. That water is gonna have a radius, and it's gonna have a height. Hey, the cone itself has a reason, Aziz. Well, it tells us that the eight radius is 45 and the height, and that tells us that eat either, really, with volume using is minus 50 meters Q per minute because it's back to the residence where the question is, how fast is the water level falling? So for part A, is looking for a D h by B t is equal to what When each is five meters. So when I want you to notice is that the ratio of radius toe height is going to be 45 26 Okay. And if you simplify 45/16 advisable for you get 15/2. But that means that radius can be written in terms of the height or the height Radio. We're trying Teoh the height by itself at this point because We're looking for the right radius as 15 h over to and that's my volume formula is 1/3 are on my are in that volume Formula 1/3. Oh, I times 15 over you. I'm speak also each. That would be something 1/3 by times and then 15 squared, which is you. 125 over you square it, which is four times each. Where demand now 225 um is divisible by four. Um, I'm sorry for I was just checking on my calculator to make sure I was saying the right thing, and I He was about three. It is by three or four. Here we go. Not a cute. Okay, that. So now I can take the derivative of that. I'm gonna get evey by DT then is equal to if I jump back here. 75 pile before state as is. And then times pH, Baird and E. We're looking for the evening. Don't anybody DS minus 50 where you get 75 or four. About three aged, which is about five. Weird. And then I th by d. K. So what? I end up with his minus 50. Is he was 75 3 a witness, 200 and 25 divided by four. And then times 5 25 What is your plan to kill you in just a second? What do you think like this? And this? Just make that two, um, 25 a d h d e When I saw this is going to become four time to so fine over 225. And that is in meter per minute. Okay, But that's my answer to part a, part B says. But about the radiant. So after where formula, which is back here, we noticed that there is 15 have. So that means d R I t is 15 half of the 85 p and so, dear, by E. T. Is going to be 15/2, multiplied by minus eight over 205 125. 15. You get 15 here and that eight divided by who is four. My answer becomes minus 4/15 and it's also in meters per minute. So the range it is striking that the height is shrinking at that rate while the water comes out and it makes sense right that the height of the radius there are going down because the water level was also going down. If you look at the picture as the shrinks, okay, as it gets closer, includes of the bottom, you're gonna smaller radius at a smaller height.

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