a-spherical-cap-the-smaller-part-of-a-sphere-cut-through-by-a-plane-has-a-volume-vfrac13-pi-h23-r-h-

Question

A spherical cap (the smaller part of a sphere cut through by a plane) has a volume $V=\frac{1}{3} \pi h^{2}(3 r-h),$ where $r$ is the radius of the sphere and $h$ is the height of the cap. What is the volume of Earth above $30^{\circ}$ north latitude $(r=6380 \mathrm{km}) ?$ (Hint: $30^{\circ}$ north latitude is $30^{\circ}$ above the plane passing through the equator measured from the center of Earth.)

Ashley High · Accepted Answer

So for a my shape looks like this. I know a equals four r equals five and find out what my height issue right here. So since this is a Pythagorean triple, we know that this side here has to equal three. And since our radius is five, that means that H has to equal to Okay, so I plug into my formula V equals pi times two divided by six times 34 squared plus two squared. So let&#x27;s simplify. Some of these two and six can be simplified, so v equals pi over three, four times 12 Ford squared is 16 times three is 48 plus four. So B equals pi over three times 52. So when I plug this part into the calculator, I get V equals 54.45 and this would be feet cubed now for part B. So for part B still going to draw my triangle here. I know that our is 34 centimeters and a is 30 centimeters. I need to find hh. So for this one, I have to do with a degree in the room. So I&#x27;m just gonna name this might be. So I have 30 squared plus B squared is equal to 34 squared as a track 30 square from both sides. So B squared equals 256 square root and be equal 16. So this is 16. So I was attract 34 minus 16 to find out what my age is, which is 18. Now I can plug it into my volume formula, so volume equals pi. Times are height which is 18 divided by six, three times a squared, which is 30 squared plus 18 squared. Okay, so I can simplify 18 and six can be simplified into three. So b equals three pie and I have 30 squared, which is 900 times three, which is 2700 and then 18 squared is 324. So I have 324 plus 2700 so v equals three pi times 2024. So I can put all of this into the calculator. So I get 28,000 500 and about 53 this would be centimeters cubed for my volume. Now for part c. Okay, so see, draw my picture again. My radius is 13 and my height is eat. So I know that this site here at the bottom has to be five. So this one is another Pythagorean triple. So I have 5 13 This last sight here has to be 12. So I can play this in to my formula, so v equals pi. Times eight divided by six, three times 12 square plus eight squared. So eight and six I can reduce. They broke computers by two. So have pi times for over two toe over three and then I have 12 square squared, which is 1 44 times three, 432 plus 64. So I have re equals four pi over three times 496. When I plug this into the calculator my volume equals 2000 and 77.64 And this is going to be meters cubed. And now for part D. So I&#x27;m still drawing the same triangle that I had. I know that my radius is 75 inches. My height is 54 inches. So if I do 75 minus 54 it&#x27;ll give me this side is 21. So buddies Pythagorean theorem to figure out what a is So 21 squared plus a squared equals 75 squared so I can subtract 75. So I&#x27;m gonna subtract 21 squared from both sides and a squared equal 5184 square root both sides in a equals 72. So this side is 72 Now I complete this into my volume formula. Some of volume equals high times my height, which is 54 divided by six, three times A, which is 72 x squared plus h, which is 54 square. I&#x27;m gonna put this into the calculator as is and be ends up Equalling. 522,000. 170 point 3398 and this is inches cute.

Sahil Patel · Answer

so we could find this fear. The flame atmosphere by using why equals square of R squared minus X word and integrating for medical arts are. But instead we&#x27;re interested in a slice over here from H to R. So take the integral Was pie from Mitch or of R Squared? Minus expert? Yes. So that would give us R squared X minus X cubed over three from R T H times pi so that gives us are huge minus R squared H minus are acute over three minus age. Huge applause. It&#x27;s cubed over three times. Hi! So we can supervise out and gets two are cute over three minus r squared age. Plus it&#x27;s cute over three times pi. That&#x27;s the volumes of saying of the segment.

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.

Andrija Isakov · Answer

and this question were given a formula for the volume of the spirit will cap, which is the smaller piece of the sphere that you get when you slice it into two parts. Here&#x27;s a spherical cap. This not a shaded. So given in this question that a spherical camp has a volume of five pi over three. And this is the formula for the volume we should find where we need to find the how the Radius R changes with the thickness of the satirical com page. And then we have to find what the value of that rate of change is when are equal to an eight equals one. Okay, let&#x27;s get started. Noticed. If the volume is five point over three, then five equals this, which is we are a squared minus H cube. Now, to find a DRD age, we need to use implicit differentiation, differentiating with respect to H. We get zero equals something which is a product rule, and I&#x27;ll leave that for later. Minus three h squared minus their square, just three x squared. And that&#x27;s the derivative of this. With respect to age and now the middle part, let&#x27;s treat three h squared as one function and R squared as the other function. So what&#x27;s the derivative of our with respect? The derivative of our times three. X squared The derivative. Remember, it&#x27;s the derivative with respect to age, so the derivative are would be this times 38 squared is three is this whole thing. And then the other term of the product rule is the derivative of three h squared, which is six h times are okay moving on. Let&#x27;s get the rth onto the left side and divide everything by three to simplify and then we can solve for New York th which equals page minus two are all over it. So now we do the second part. What is the value of DRD H? When R equals two and H equals one? Let&#x27;s quickly do that. That should be an easy substitution, and we got at the value is negative. Three

Mike Gaerlan · Answer

So let&#x27;s plug in our volume first here into this equation. So you have five. Pi over three is equal to pi. Age squared times three R minus H divided by three. So the divided by three skin council out and so can the pie. So we&#x27;re just gonna be left with five on the left hand side. So we have five is equal to H squared times three R minus H. Now we&#x27;re going to take the derivative with respect to H on both sides. We&#x27;re gonna take d d h of five is going to be able to d d h of and I&#x27;m gonna go ahead and multiply this h squared on both of these terms. So we&#x27;re gonna have three our age squared minus h cute like so Now the derivative of this left hand side is the derivative of a constant, so that&#x27;s just gonna be able to zero we have zero is equal to. And then now the derivative with respect to H here, we&#x27;re going to need to use product will, so we&#x27;ll do first. So that&#x27;s three are times the derivative of the second. The rivet of this respect age is just to H and the now plus the derivative. Uh, well, do second just h squared times the derivative of first, which is going to be times three and then d r d h because of we&#x27;re just taking the derivative are, which is the rth men minus than the derivative of this with respect to each is gonna be just three age squared. Next, we&#x27;re gonna move all of our non p h d rth terms. We&#x27;re gonna move them over to the other side. So we&#x27;re gonna have three h squared and also notice before I do that. Um, the there&#x27;s a three on all of the terms. 33 and three squeaking. Counsel those out. So again, now we&#x27;re gonna have eight squared and then minus two r h design of equal to h squared d r d h. So then we can divide by an a channel on all the terms. So we get that d r th Yeah, All right. This in black. That&#x27;s the r d h is going to be able to. And again, we also have an h um, on all the terms we can divide, we can cancel out that age so we&#x27;ll have just h minus. Two are all over each. So now we need to evaluate. I&#x27;ll wear de rth whenever we have. Our is equal to two and H is equal to one. So d r D h at one comma too. This is equal to will have one minus two times two all over one. So you have that the rth at one comma too. This is evil to we have one minus four, which is negative. 3/1. So that&#x27;s just gonna be able to negative three.

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A planner for an electronics supplier plots a gri…

Problem 39 Easy Difficulty

Answer

$1.70 \times 10^{11} \mathrm{km}^{3}$

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Allyn J. Washington, Richard S. Evans

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$1.70 \times 10^{11} \mathrm{km}^{3}$

Basic Technical Mathematics with Calculus

Anna Marie V.

Samuel H.

Michael J.

Joseph L.

Vectors Intro

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

Anna Marie V.

Samuel H.

Michael J.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus