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  • Hey, Vsauce! Michael here! If you core a sphere; that is, remove a cylinder from it, you'll be left with a shape

  • called a Napkin ring because, well, it looks like a napkin ring!

  • It's a bizarre shape because if two Napkin rings have the same height, well

  • they'll have the same volume regardless of the size of the spheres they came from! (Cool)

  • This means that if you cut equally tall napkin rings from an orange and from the Earth,

  • well, one could be held in your hand. The other would have the circumference of our entire planet,

  • but both would have the same volume...

  • I mentioned this counterintuitive fact while making a Kendama with Adam Savage. Check that video out if you haven't, or better yet,

  • just come see us. We're bringing brain candy live to 24 new cities this fall. It's going to be busy, but right now

  • We're talking about balls and coring them!

  • I have here TWO napkin rings from very differently sized spheres; one is from a tiny ball,

  • just a little tomato that I've cored, so it's got a little hole in it right there.

  • The other Napkin ring is made from an orange,

  • but both Napkin rings have the same height. The tomato has a smaller

  • circumference than the orange which means less volume, but its ring is thicker which means more volume. Both of those effects

  • exactly cancel out. So these two napkin rings have identical volumes they take up the same amount of space

  • By the way orange oil is flammable

  • To see why the napkin ring problem is true,

  • let's discuss Cavalieri's principle. It states that for any two solids like these two cylinders I've built here

  • sandwiched between parallel planes if any other parallel plane

  • Intersects both in regions of equal area no matter where it's taken from

  • Well then the solids have the same volume. That's clearly true here these cylinders are built out of stacks of VSauce stickers

  • 100 in each Stack so their volumes are the same

  • if I

  • Skew one of them like this it shape will change, but it's volume hasn't it's still contained the same amount of stuff

  • I haven't added or subtracted stickers

  • And Cavalieri's principle ensures that they still have the same volume because any cross-section taken from down here up here in the middle anywhere

  • Will always give us a region of the same area as the other because those regions are always equal area circles.

  • Now let's Apply Cavalieri's principle to Napkin Rings

  • We can see that two napkin Rings with similar Heights have

  • identical volumes by showing that when cut by a plane the area of one's cross section

  • Always equals the area of the others now to do this

  • Notice that the area of the spheres cross section minus the area of the cylinders cross section gives us the area of the Napkin rings

  • Cross section. Depending on where we slice the Napkin ring the cross sections will have different areas

  • But they will always be the same as each other. Let's calculate the areas of these blue rings

  • first of all let's call the height of the Napkin ring h and the radius of the sphere they're cut from

  • capital R

  • Alright, perfect now a cross section of a sphere like this and a cross section of a cylinder like this are both circles

  • So their areas can be determined by using Pi times the radius squared

  • So if we want to find the area of the spheres cross section and subtract the area of the cylinders cross section

  • (I'll draw a picture of a cylinder here), all we need to do is take Pi

  • multiply it by the radius of the sphere cross section

  • Square that and then subtract Pi times the radius of the cylinder Squared, but what are their Radii?

  • Well, if this is the center of the sphere we can draw a line straight up to the corner of the cylinder down the side

  • of the cylinder and then connect to form a right triangle

  • The Pythagorean Theorem will really help us here it tells us that the length of one side squared plus the length of the other side

  • squared equals the length of the hypotenuse squared

  • Now this distance right here this side of the triangle what we want

  • It's the radius of the cylinder, so we'll call this the little r radius of the cylinder (beautiful little picture there)

  • so the radius of

  • the cylinder

  • squared plus

  • this side length, which is just half the height of the cylinder, so the height of the cylinder divided by 2

  • squared

  • Equals the hypotenuse squared the hypotenuse happens to be the radius of the sphere itself which is capital R

  • Perfect now let's solve for the radius of the cylinder. Which is what we want

  • We'll just subtract h over 2 squared from both sides that'll give us the radius of the cylinder

  • squared

  • equaling the radius of the sphere squared minus

  • Half the height of the cylinder squared

  • We can take the square root of both sides so that we wind up with the radius of the cylinder

  • equaling the square root of the radius of the sphere

  • minus 1/2 the height of the cylinder squared

  • Perfect ok now let's take a look at the area of a cross-section of the sphere now for this

  • let's draw a straight line from the center out to the edge of

  • the sphere's

  • Cross-section, and we'll go straight down and connect back up, hey look! Another right triangle

  • let's call this height y

  • And notice that this distance now the side of the triangle down here is actually the radius

  • of the circle

  • Cross-section up here. They're both equal so we even want to solve for this the radius of the circle that is the spheres cross-section

  • Okay

  • so we know that the radius of

  • The sphere's Cross-Section squared plus this distance squared (which is y)

  • Equals the hypotenuse squared well, what do you know the hypotenuse is the radius of the sphere again (capital R)

  • Ok let's subtract y squared from both sides the radius of the spheres cross-section squared

  • equals the radius of the sphere squared minus

  • Y squared will take the square root of both sides and end up

  • learning that the radius of the sphere's cross-section equals the square root of

  • The radius of the sphere squared minus y squared. Y is the height that this

  • Cross-section is taken from above the equator the higher up we take these cross sections of the sphere the smaller their radii will be

  • Whereas the cylinders radius is always the same no matter where we cut from

  • Anyway, let's take these two Radii and plug them into our formula

  • Okay, the area of the cross section of the sphere is what we want first

  • Okay, that's just the square root of R squared minus y squared

  • Not too bad now the radius of the cylinder is the square root of R squared minus

  • Half the height of the cylinder squared now what you might notice is that we're taking the square root of something and then squaring it

  • These actually cancel each other out

  • perfect! Much more simple looking

  • But now let's distribute Pi to the terms inside the parentheses so pi times R squared gives us Pi R

  • Squared Pi times negative y squared gives us negative pi y squared

  • Then a negative Pi times R

  • squared is negative pi R squared negative Pi times negative h over 2 squared is positive Pi

  • h over 2

  • Squared. Great, now we can keep simplifying but what you might notice is that we have a pi r squared and a

  • minus Pi R squared, well, that equals 0 so these

  • Completely cancel each other out, but what we're left with are

  • Terms containing no mention of the spheres Radius whether the radius is large or small

  • Doesn't matter all you need to know to find the area of the cross section of a napkin ring

  • is the height of the Napkin Ring

  • Y, of course, is bounded by the height of a napkin ring these blue areas have the same area as each other and this will

  • Be true no matter where we cut the cross section across the napkin ring meaning by Cavalieri's principle that both Napkin Rings have the same

  • Volume

  • Yay :3

  • (with teeth)

  • But what is this mean for you for life in the universe?

  • Well as we know if you like it you should put a ring on it

  • but if you like It,

  • don't know it's finger width and only want to offer it a predetermined amount of material

  • you should put a napkin ring on it.

  • And as always

  • Thanks for watching :D

  • On August 21st

  • 2017 there will be a total solar eclipse the shadow of the Moon will race across the Contiguous United States

  • It's going to be incredible and a little bit scary

  • I'm sure I will be viewing it from Oregon with my friends at Atlas, Obscura. I can't wait, but keep your eyes

  • Safe if you want to view the eclipse you have to have special eye protection

  • the curiosity Box comes with such glasses these block

  • 99.999% of visible light that's what it takes to be able to look right at the sun as actually what I love about these glasses

  • There's no eclipse going on. You can still just look at the sun

  • Notice that it's a ball. Maybe imagine what kind of Napkin ring

  • You'd like to make it into. The current

  • Curiosity box is my favorite. The one that you'll get if you subscribe right now comes with all kinds of cool stuff that comes with

  • A poster showing that all the planets and pluto can fit between the earth and the moon it also comes with science gadgets like these

  • levitating magnetic rings

  • pretty cool also a portion of all proceeds go to alzheimer's research, so it's good for your brain and

  • Everyone else's brain check it out. I hope to see you at Brain Candy live and as always

  • Thanks for watching

Hey, Vsauce! Michael here! If you core a sphere; that is, remove a cylinder from it, you'll be left with a shape

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B2 中上級

ナプキンリング問題 (The Napkin Ring Problem)

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    林宜悉 に公開 2021 年 01 月 14 日
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