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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.)

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

Calculus 3

Chapter 29

Partial Derivatives and Double Integrals

Section 1

Functions of Two Variables

Partial Derivatives

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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'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'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'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'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'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'm gonna put this into the calculator as is and be ends up Equalling. 522,000. 170 point 3398 and this is inches cute.

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