determine-the-required-values-by-using-newtons-method-a-dome-in-the-shape-of-a-spherical-segment-i-2

Question

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. $24.12 .\left[V=\frac{1}{6} \pi h\left(h^{2}+3 r^{2}\right) .\right]$

Robert Leedy · Accepted Answer

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&#x27;re perpendicular to the y axis but our bounds here about X when we&#x27;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&#x27;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&#x27;re gonna plug in negative 60. The negative 60 scored over 60. That&#x27;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&#x27;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&#x27;re going to multiply everything by 60. So we get 60. Why close? 60 squared and we&#x27;re going to move. Our negatives were going to actually make everything negative here. And then we&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s really a 1/2 power. So we&#x27;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&#x27;re going to plug in 60 squared and plug in 60 square, but we&#x27;re subtracting those, so we really just get a volume of zero.

Amy Jiang · Answer

Okay, so eyeballing me is just gonna go from our minus h to r of pi times R squared which is a radius My ex wife Word. Do you? Why thank this gives me hi time r squared times one when it&#x27;s why cubed over three evaluated at arm and our mind It&#x27;s h Okay If you simplify this you get high times are eight squared wise one over three cubes We know that our is equal are big are people too Why times r squared for spit squared over to H squared minus one over three inch cube If you simplify those two yet quiet h over six times three r squared plus squared.

Amrita Bhasin · Answer

this question asked us to determine the function described and then use it to answer the question given V is pi r squared h. What we know is that we need to write this just in terms of our so we know are in this context is gonna be the square root of B over pi h. The reason there&#x27;s a square root is because you take the route of both sides in order to get rid of the R squared. Because remember writing this just in terms of our now what they&#x27;ve told us is that we know we need to consider a volume of 300 cubic meters, which means plug and 300 and then the age is gonna be six feet and this gives us the square root of 50 divide by pi meters.

Maya Mysore · Answer

So this producers question, which is The calls are high, each squared well spotted Third Hi. Each cute and they domesticated in terms of our So we can do some stuff Leader with the surface every question. And so in order to do that this do I buy slow it one plus 1/3 by age cute We still have our pie each square on the site and then we divide both sides by y h squared cancels out there, you pie each squared And now this part gives us the equation for our And so we can use that in the next part with surface area So surface Siri is you or it we can plug in that are here This equals two pi times the close wonder h cute over why each squared And so this cancels partially we get that pie HL we get that out, we get one each out but Liza&#x27;s with V plus 2/3 hi h cube over each And so when they ask us to plug in the value of by wonderful V, we can plug it in and we get one variable left in the equation and that gives us something that we can actually grab. It&#x27;s only right that just to make it easier to see, but two socks 500 is 1000 plus 2/3 pi h sward. Um, and I&#x27;m starting to buy, but 1000 trips simply by H. Cause I already took care of each squared turn, bringing it down from each cute. And so when you graph it, um, you can use a graphing calculator to do this to check. You&#x27;re gonna get that vertical else Internet zero. It&#x27;s gonna look roughly like this. Oh, and then you&#x27;re gonna get a minimum. H what you could, Teoh Peters. And then, if you want to find out what the or value is, you&#x27;re gonna plump that back into the original question is equal to plus over squared. So you plug that. Therefore, is this still 100? And you have been a calculator math. But I&#x27;ll just tell you, it comes out to about 6.2 centimeters

Donald Yeh · Answer

Okay, so we want, um, using double in a girl&#x27;s derive. Uh, the form lift for the volume of a right circular cone. So right. Circular cone looks like So, um, basically circle on the bottom with radius A in this case and then height H and we want to find the volume with us using double in your goals. Uh, and in order to do so, uh, we&#x27;re actually going to need to do or need to change a few things about this or actually rewrite this in a way that is a lot easier to software. Um, so we&#x27;re actually gonna use the hint given to us in this problem is basically the fact that we need to show that this volume right here, given inside here, is actually equal to the volume under Z equals age and above, so below. And then above Z equals h over a square root of X squared, plus y sward. So let&#x27;s just draw this out. So is equals. H is somewhere up here. Just say horizontal plane, and then we just need a draw. This and this basically just, um the square both sides. C square is equal to h squared over a squared of the quantity Explorer plus y squared, which is just basically, um, some factor multiplying this and this is basically this entire thing is just a cone. So what we can draw is a cone extending from the origin and only the top half. Because in the original function, a Z is only your fitted to be the positive component because there&#x27;s a square root. So square is gonna leave deposit values and H and they are always positive. So this should always be a positive value altogether. Um, but we need to know what the intersection between Z equals h and the surfaces in order to verify that this is the same as this. So we have h is equal to say them equal to each other. We have ages ago to H over a time squared of X squared plus y squared. They just canceled because we know H is hopefully not to be zero in this case. So division by age is all right. Um, I just always have to be weary of division by zero, because that is a big no no. In mathematics, multiply both sides by Amy at the this than a square is able t x square plus applies with the square both sides. This is basically just the circle centered at the origin of Radius A, which is basically the same as this. So that part checks out. Yes, we have radius a here. And since Z is able to age basically from here, all the way up to here is the height of age. So that checks out as well. And since cones are exclusively or at least right, um, right. Circular cone xrx loosely described by their height and radius. And since their high ingredients are the same that these two are that same cones, OK, so we just need to find the volume in here. Um, in order to do so, we&#x27;re actually gonna have to It&#x27;s gonna be a little tricky, because if you just did the double integral of this function right here for this, uh, region, we would actually find the volume here. And that&#x27;s what we what we want. We actually want to find the volume here, right? So we&#x27;re gonna have to basically find the volume of this entire cylinder and then subtract out the volume found here, which is pretty simple, but I&#x27;m just gonna write this in terms of all of the double integral, because that&#x27;s a lot more generalized and realizing that this is a cylinder like perhaps maybe this isn&#x27;t a cylinder and we want to actually find balling with it. So let&#x27;s do that instead. And I&#x27;m gonna write this double unusual in terms of polar coordinates because everything is very circular and dust. This surface is also X squared is in terms that I swear plus y squared, which translates toe polar quite nicely. Um, basically, we need to find the volume of this cylinder so the cylinder would just be over this region are so we can say that the surface is just age because the easy old age and then minus um basically this surfaced because then that would find the volume here. So we have ah h over a and then screw of r squared. And all of this is mostly almost played by our DRD data. And then we need to find the limits. Eso now our region art is the circle with radius a certain at the origin. So it looks like this. So our ranges from zero to a and then data ranges from 0 to 2 pi. It is important to realize that a in a trip of constant values in this case, they&#x27;re not variables, because for any specified Cone h and they are gonna be some new value, so they are not variables in this problem. Uh, let&#x27;s first just some five. It&#x27;s a little bit. We have aged mice ate over a times are because our square toe ah, spirit of that is just art Artie. Artie Fada. Then we just stumbled. Five are out hr my h over a r squared d already data and will notice that the entire inside integral here is just in terms of our so with respect to the data angle on the outside, this is just a constant values. Who can just slide decide to the front. There&#x27;s a of a charm as h a r squared d r times 0 to 2 pi of one d data and that we could just sell for these separately. We have h r squared over to my h r. 2/3 over three a from zero to a times two pi because this England, a girl of function one is just the length of the interval. Uh, this is equal to H A squared over two, just blinking a for our and then h h the third over three a and then secured what we get when you plug in zero. But since each has a factor of our or multiple factors of are in the form of R squared arch of Third, that means that each component is just gonna be multiplied by zero. So that means that the subtraction this is gonna be a zero and the most like this by two pi secrets h a squared over to my h a squared over three because I just can&#x27;t have aces. And A is always with me non zero in this case, hopefully or else the volume would be trivial toe calculate for So the cancellation of aces All right, in this case that we have three a. J squared over six minus to a J squared over six multiplied by two pi. This means that we just have 1/6 of h a squared times too high. This is equal to, um hi. A squared age over three and that would be our final answer. And This is very close to, um, the volume of our rights. You click on that you may have memorized, which is just 1/3 pi r squared age where the H is just match up and then our and a matchup because a is what are basically the radius of the cone in this case.

Problem

Determine the required values by using Newton's m…

Problem 29 Easy Difficulty

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus