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Solve the problems in related rates.Coffee is draining through a conical filter into a coffee pot at the rate of $18.0 \mathrm{cm} .^{3} / \mathrm{min}$. If the filter is $15.0 \mathrm{cm}$ in diameter and $15.0 \mathrm{cm}$ deep, how fast is the level of coffee in the filter changing when the depth is $10.0 \mathrm{cm} ?$ (Hint: Use $V=\frac{1}{3} \pi r^{2} h .$ )

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 4

Related Rates

Derivatives

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the question gives us this picture. Um, where you have a coffee pot down here and water is training through a filter and into the coffee pot. The conflict fight is a cylinder, and the coffee filter is a cone. Okay? And it tells us the rate in which the volume is flowing. The raid was the coffee. It's flowing this 10 cubic inches. Her minute. Okay. And then it tells me that the height of the car you may call E. It wants to know what's going on with the height when the height, it's five. What is the three of change of the height? It's and okay, the question is pretty easy because we know the radius A The radius is gonna be three here, and it's going to stay as three inches that when we think about the volume of a cylinder, the volume cylinder is pi r squared each, but we know the volume is gonna then become pie where I know the radius is three. So that means the volume is nine pi h. Let me is d V by d. T. Then is just evil. The nine pi times th by t t and since we know D V by D T is 10. We're gonna get hand is equal to nine high times th by d t if we jump back here. So what we have to do is divide, so we get 10/9 pie. That's the rate at which the height is changing. Okay. And it's in inches at her minute. Okay, so that's my d h by the teen out part B says what's going on in the cone. All right, so if you look at the cone uh, this is the radius the cone. This is the height of the cold. Um, and we go back to my what's Rio? Then we go back. The volume that's in the code is to be 1/3 pi r squared. Okay. However, if we look at this, we know that the radius is changing. Bus is changing, and at a rate such that the ratio here, this is three right over six. The 3/6 is equal to are over age, which means that h over to this reduces the 1/2 is equal toe are so what we can do If you look at it, I want to know how that is a little cold falling. So that means that wants to know about each. So what I'm gonna do is I'm gonna actually say deep are not DVD body t just yet. Just V is 1/3 party times age over to where I'm so if V is equal to that becomes 1/3 pi times each squared over four times each or just fly over 12 each human. Okay, so if I take the derivative of that I get d V by d. T is equal to and then it's gonna be three pi over 12. H swear times. The eight by the team. Hey, so assuming that the height is also five inches in the cone at that point In other words, we assume that the hi is the same here as it is down here. Although it it's actually it doesn't tell us. Oh, sorry does tell us that tells when the in the Conus five inches. Okay, we may have to redo that way. We don't have a read of this because it didn't come into play. So yes, we're allowed to use the five inches. Okay, good. So we can just go then three pi over 12 which is actually gonna be five before. But then times five squared and sometimes th by t. And that's equal to In this case, it's 10 but it's minus 10 because it's flowing out. So D h by d t if I saw that is going to go minus 10. Okay, times 12 divided by three pie 35 If I reduce that. Basically ideas, brother. 12 up. Okay, the three pie down in the five squared is 25 down over. That makes sense, but I just divide these by five. That becomes, too. And after game five, and then divide these by three. So that becomes forward and one So D h by the tea, then his minus eat over five pi and it's again gonna be in inches for minutes.

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