Transcription
Hey, Miss Mlos here. And, um, in this lecture, we are talking about uses of conic sections. Um, so it's going to be pretty quick, but, um, I'm just going to kind of explore some of the ways that these conic sections are used in real life. And we're not going to talk about circles at all, 'cause I think you guys have known about circles since you were like two years old, and I think that's pretty evident to us. So, we're going to start with parabolas.
So, um, the whole reason why a parabola is often used, um, in real life is because with a parabolic shape, any line is reflected to the focus. So, the first place that we often see, um, parabolas used is in a satellite dish. Here's a picture of our Direct TV satellite dish, and you guys can kind of see how this is parabolic in nature. And the reason why they use this shape is because all of the rays will come in that have the signal, and it will be reflected from the parabola into the focus, which is what is actually transmitting that signal to your TV. Okay, so a satellite dish, something we see quite often, is a parabola.
Another place where we see parabolas used quite often are in car headlights, flashlights, spotlights, and telescopes. And if you guys have ever wondered, um, how a car headlight works, okay, obviously here's a picture for a car headlight, but if we look, there's actually a parabolic reflector that is here, and the light bulb, instead of going out, it actually faces in, and it's reflected off of this parabola. Okay, so the light bulb is at the focus, it's reflected off of the parabola, so then everything comes out straight. Okay, so that's what causes, um, a headlight to take up all this space when it actually reflects all that light. Okay, so this parabolic mirror is also used in spotlights and telescopes and flashlights.
Another use of the parabola, um, we actually see it in what we call a sonic boom. Okay, so if we look here, here we go, just for you guys, um, this shape is parabolic in nature. So, when the plane breaks through the sound barrier, the noise that is heard actually travels here in this parabolic shape, okay, which then emits that sound everywhere on this particular shape. Okay.
Another place we see parabolas every day is whenever objects are thrown in the air. And I think often this is most evident, um, with something as simple as like a little boy throwing a baseball up into the air. When he throws the baseball up into the air, it's not going to come straight back down, but there will be some parabolic shape in it. Okay. In the midst of March Madness right now, we definitely know when someone is shooting a basketball that looks very parabolic. Okay. If you are, um, serving in volleyball and you're tossing the ball up, you know that it doesn't come straight down; it definitely reacts, um, in a parabolic manner. Football, even though it appears to be straight, it's not quite straight; it is parabolic as well. This is also true when things like bullets, arrows, or missiles are shot. Okay. We can kind of see in this picture the bullet trajectory. There we go. The bullet trajectory is not a straight line; it definitely is parabolic. So, if you have ever shot a gun, if you have ever been throwing darts or something like that, you can definitely know that you don't aim straight for the bullseye; you do need to adjust slightly. Okay. And they'll actually use this if you guys watch CSI, um, sometimes, um, in the episodes, and if they're trying to figure out where a bullet came from, they will actually like plot out the trajectory of the bullet and use this parabolic motion and this parabolic shape to determine where the source of the bullet was. So, a parabola has definitely seen a ton in real life.
The final application that we're going to talk about parabolas in, and keep in mind there's way more, but, um, the fact that parabolas are actually very structurally sound, um, means that a lot of architects will use them in the building of bridges and other sorts of buildings. So, here are some examples, um, of some bridges that have parabolas in them. And I never really thought about this before, but now I'm totally like a math nerd, and I think about parabolas whenever I see a bridge. But we can definitely see this is a very common bridge shape where the parabolas are on the under, on the like bottom side, the underneath part. And we can see it in these two pictures, um, here, like with the Golden Gate Bridge, they actually use like the suspension is in a parabolic shape to help distribute the weight. We also see them in a lot of buildings, like here's the Arc de Triomphe, and we can see that we have the parabola shape in there. And here's another picture of a random building, but we can see that there are a ton of archways which are all parabolic in nature. Okay. So, um, if you go through and look at, um, just life around you, you are destined to find a ton of parabolas, 'cause they are very, very commonly used.
Ellipses are not used quite as often as parabolas, but, um, the way God designed the universe, the universe, um, he used ellipses as the planetary orbits. And we can kind of see that demonstrated in this picture over here, where we can see each, um, planet kind of circling. And obviously this is an old picture, 'cause it counts Pluto as a planet. But regardless, um, with an ellipse, um, believe it or not, they used to believe that the planetary orbit was circular in nature. And the scientist Johannes Kepler, um, back in the early 1600s, in fact in 1609, he published his first two laws about planetary motion. And, um, he analyzed the astronomical observations of Tycho Brahe. And what he really found was that every single, um, like planetary orbit is in the shape of an ellipse, and the Sun is at one of the two foci. And to be quite honest, I'm not sure what is in the other focus, which is kind of weird. But this was a very radical view, and, um, often it was viewed as a heretical view, um, because it was believed that the Earth and the Sun were like the center of the universe. And Copernicus was kind of the leading expert in this, and his theory was that the Sun was the center of the universe, and everything was circular around that. And this kind of opened up the door to what we now know in terms of like astronomy, um, because of these elliptical orbits.
Another place we will see ellipses is, um, there are some elliptical theaters and galleries that, that obviously in the title, are shaped as an ellipse. And the whole purpose of this is that the definition of an ellipse: it is the sum of the distances from the two foci. So, if we have two foci, wherever they may be, um, if you're actually standing at one focus, you can hear everything going on at the other focus, even if it is completely across the room. In fact, um, some of you guys may have, may watch How I Met Your Mother, and there was an episode on How I Met Your Mother where they're in, um, like an elliptical room, and this very thing happens where if you stand in one place, you can hear everything that's happening somewhere else. Okay, so that's another use of an ellipse.
So, now let's move on to hyperbolas. Um, mathematically, the reason why a hyperbola is nice is because it gives us a maximum volume with a minimum surface area. So, one of the places we see this, um, around the world is in the use of building nuclear cooling towers. So, looking at this shape, okay, we can definitely see that these are hyperbolic in nature, and it can hold a maximum amount of material inside with a minimum amount of surface area, which, um, allows them to keep it at the correct temperature.
Another place we see hyperbolas are actually in a water bottle. And, um, you guys may not remember this, but water bottles were not always hyperbolic in shape. And when they came out with this, um, new model, the big thing was that it was made with less plastic. Okay, so it's using less material. And the whole reason is because of the hyperbolic shape: it holds the same amount of water but uses less material, so it is better for the environment.
And here's the final example we're going to talk about, but a hyperbola is also used in a trumpet. So, what this does, it actually maximizes the amount of sound that comes out of such a small portion of that instrument. And, um, you guys have probably put your hands together to talk really loud, which amplifies the sound. In case you didn't get it, I just did that while I was, I demonstrated while I talked about it. But anyways, um, conic sections are definitely used all over the place in real life, and, um, I'm excited to see what uses of conic section you guys find for your responsive writing.