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Determine the required values by using Newton's method.A dome in the shape of a spherical segment is to be placed over the top of a sports stadium. If the radius $r$ of the dome is to be $60.0 \mathrm{m}$ and the volume $V$ within the dome is $180,000 \mathrm{m}^{3},$ find the height $h$ of the dome. See Fig. $24.12 .\left[V=\frac{1}{6} \pi h\left(h^{2}+3 r^{2}\right) .\right]$

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 2

Newton's Method for Solving Equations

Derivatives

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All right, We need to find our volume. But we also have to set up the integral here are not given exactly what we need. We are told that we're perpendicular to the y axis but our bounds here about X when we're perpendicular to the y axis we need to use from a to B as a function of why do you Why? So we can't just plug this in. We need to first get it in terms of why. Which means we need to solve for X. But it also means we need to change our bounds for wide. In this case is pretty easy, because everything here is 60. So we changed our bounds. So are a why, and r b y are going to be for a We're gonna plug in negative 60. The negative 60 scored over 60. That's simply gonna be zero. And for B plug in 60 we get 60 squared over 60 minus 60 minus 60 is again zero. Now, getting this into terms of why we're going to move everything away from the X and then take the square root. So we will add 60. So we get why plus 60 We're going to multiply everything by 60. So we get 60. Why close? 60 squared and we're going to move. Our negatives were going to actually make everything negative here. And then we'll take the square root of everything, and that was going to be equal to X. Now it looks like a pretty annoying problem. But the 60 squared. We're gonna leave it written A 60 scored. This is a very large number. However, it makes things pretty usable. Now, one thing we run into when our bounds about zero. So we're going to go from 0 to 0. Negative. 60. Why? Plus 60 squared and the square root. We use u substitution here and we are going to sign. Do you? Why use U substitution. So we get u is equal to negative 60. Why? Plus 60 square Do you is going to be able to negative 60. Do you, uh, do Why? So that means de y is equal to you know Why do you Over Negative 60. All right, so we're going to plug in the square. You here. So we have we also you plug in our zeros, but you get zero plus 60 says is just 60 scored in 60 squared. All right, solving this weaken were negative. 60 out front, Get one over. Negative. 60 from 60. Squared to 60 squared. Square it of you, do you? That's really a 1/2 power. So we're going to increase our power by one divide by the new power it gives us but one over negative. 60 times to over three. You from 60 squared to 60 squared. Now the issue is we're going to plug in 60 squared and plug in 60 square, but we're subtracting those, so we really just get a volume of zero.

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