In problem 22. We once again have related rates once again dealing with a cone, and that means they're going to be dealing with a volume of a cone. Specifically, we're gonna be using that same equation from the last couple problems. This case for solving for the height as the time is changing, given all the rest of the information we need. But before he could go ahead and take the derivative of V. With respect to time, we need to get this down to one variable. You need to find the relationship between R and H. And since we're trying to find a HR, ready T means we want to eliminate the R value. Well, we're given some constraints. You We are told that for this entire cone that the base is 10 ft in the height is 12 ft and census nice triangle is always gonna have that same ratio. So if I split it in half for him, it's gonna be dealing with half of the base, which is gonna be the radius and at having 5 ft. And then I'm gonna have 12 ft, and that's gonna have the same ratio, as are H or whenever I'm dealing with eventually when my height is 8 ft and then I'm just gonna have, um some are at eight, but And first, what I want to do is I'm trying to solve for R in terms of H. And I could do that because this ratio is the same. So 5/12 is gonna be equal to our over h are is going to be equal to five wealth each. Now, what if I wanted to solve this? When H is equal to 8 ft? Well, our of eight going to be equal to 5. 12 times eight, which is going to be equal to then go ahead and I can cross it out. Make that two gonna become three, going to end up with 13th. But now I have all the relevant information in order to complete this problem. But I can plug in that our value into my original equation. So I have one third. Hi. I'm gonna have have 5/12 h squared. That's going to be times H that's going to be equal to. I'm gonna have 25 that's going to be over 1 44 times. That's going to be of 400 32 Gonna be pi times h Q So I can go ahead and take the derivative so D v d over DT gonna be equal to 25 over 432 pi It's just the power rule, So it's gonna be three times h squared. And remember, I'm taking this with respect. T so d h over deep. I'm just going to simplify this as little as I can. Basically, that three can just cancel out in the denominator here, and that's going to turn back into 144. I'm going to get that. This is going to be equal to 25 over 144 pi h squared th over the team. As you can imagine, this isn't gonna end up being a very pretty number. So TV over DT is 10 going to be equal to? I have 25 over 144. I'm spy. That's going to be times eight squared. That's going to be times D h over d team. But I can solve for d h over DT I just dividing you side by all those constants. So I mean, I have 64 times, 25 times pi divided by 144 and divide 10 by that number, which is going to give me that th over DT gonna be equal to 0.2. I'll do it to three significant figures Or 286 And for this, I want to make sure I keep my units And in this case, in the beginning, as dealing with feet cubed per minute and I went down to just for a minute.