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College Algebra: Apply Ellipse - Satellites

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About this Lesson

  • Type: Video Tutorial
  • Length: 8:09
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 88 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Trigonometry: Full Course (152 lessons, $148.50)
Trigonometry: Conic Sections (12 lessons, $26.73)
Trigonometry: Ellipses (3 lessons, $4.95)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/collegealgebra. The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Let's just consider ellipses that are censored at the origin, so we have that h and k business, subtracting the x and the y. So then the formula, I'll remind you, is just + = 1, where whatever value is bigger, either a or b, that will tell you who is the major axis. If a is the bigger value, then this will be a major axis this way, you'll get a shape like this. And, in fact, therefore the foci will actually be on the x-axis. If, in fact, b is larger than a, then the ellipse is going to have this general shape, and then the foci are actually going to be on the y-axis.
Let's just suppose that this is the major axis here. If this is the major axis, if a > b, then - in fact, I'll tell you where the focus is located. The foci are at + c, 0. So there are two points there: there's c,0 and -c,0, but what's c? Well, where c^2 = a^2 - b^2. So if you want to find c, you've got to take the square root of both sides. So you take this number and subtract that number and take the square root, and that will give you c, and you take a negative, which is -c. So that's the easy way of finding the foci. It's not a big deal at all. And, of course, if this were the major arc, if this were larger, then everything would be flipped around. I would have 0 + c. These would flip, because it would now be on the y-axis. And, in fact, this would change, too. c^2 = b^2 - a^2; I take the larger always and subtract the smaller.
All right, now actually this can be used to do stuff, because it turns out that when you think about satellites and orbits, those are actually ellipticals. Ellipses aren't just some sort of weird abstract thing. Suppose that we go out into outer space and take a look. Here's the earth, and suppose that we have some sort of satellite that we shove up in space. In fact, I'll do the whole thing for you live right here. So you have this satellite here and we shove it out in space. What happens is it starts to go into an orbit. And that orbit could be an elliptical one, where, in fact, the center of the earth is one of the foci. So you have some sort of ellipse that this thing is following and it goes like this. It sort of orbits this way. It gets close to the earth and it gets far from the earth, and it gets close to the earth and far from the earth. Well what if I gave you some information about this? What can you tell me about this orbit?
Well, let me tell you some information. So, in fact, what we know is that the coordinates in miles for the orbit of this artificial satellite - in fact, this one I'm thinking of is Explorer VII, can be described by the regular equation for an ellipse, + = 1, where here's what a is, a is the value 4465 miles and b is 4462 miles. Now, if we assume that the earth is actually located at one of the foci of this elliptical orbit, then when I want to know is what is the maximum and what is the minimum distance between the satellite and the very center of the earth? So let me actually now try to draw a picture of what's going on here.
What we have here is an ellipse. And this is the a value, which means this is where we're going to cross the x-axis. Notice it's a teeny bit bigger than this, so I'm going to maybe exaggerate it - 4465. This is going to be -4465, this is going to be 4462. So it's almost a circle, but not quite. And then I have minus down here. And if you draw in the orbit, it looks like this.
Now, there are two foci, one over here and then some other one over here. Well, if the satellite is sort of traveling along this path and we're told that one of them is earth - let's suppose that this focus right here is actually earth, then what I'm interested in is figuring out when is it as close to the center of the earth, when is it the closest this gets to the center of the earth, what's that distance, and the furthest? Well, the closest is plainly right here, whatever that distance is, and the furthest is actually this distance right here.
So what I've got to find, first of all, is the equation for this particular ellipse. Well, the equation for the ellipse is real easy to find, there it is. So, in fact, the equation can be given very quickly just by plugging this in, so I see + = 1. So there's the formula for this particular ellipse. Not a problem. But now I've got to find the location of the foci. Well, how do I find the foci? Remember I told you that c^2 = a^2 - b^2. So what's a^2 - b^2? Well, you have to take that number and square it and then subtract that number squared. And you can use a calculator and you'd find out that c^2 = 26781. Just that number squared, a^2, minus b^2. Now, if I take square roots of both sides, what I would see is that c would equal, roughly speaking, 163.64 miles. That is c, and that is, of course, this location right here. So this distance right here is actually 163.64 miles. And similarly, this distance right here is the same, 163.64 miles.
Okay, so what I want to find out is this distance here. How can I found out that distance? Well, let's see, if I know this whole thing is, in length, 4465, if I know this is 163.64, I can actually find this length just by subtracting. And if I subtract, I see that this length right there is 4301.36 something. So that's how many miles, 4301 miles, basically, is the closest the satellite gets to the center of the earth.
Now, where's the farthest? Well, the farthest is way, way out over here. How can I find that? Well, I've got to look for this distance right here. Well, how do I find that distance? Well, I know that this distance right here was given to us to be 4465. So all I have to do is take that and add on this, and this we already saw was 163.64. So if I add 163.64 to this, which is 4465, I would see that this length right here, this big length, would be 4628.64. And so, in fact, that represents the furthest this satellite is from the earth. Notice these numbers are actually pretty close together, and that's because this thing is very close to being a circle. Notice, in fact, the major axis and minor axis are almost the same length. So it's not surprising, but it's interesting. You can actually figure these things out just by knowing that we are in an elliptical orbit and we can find out the location of the foci.
Okay, see you soon.
Conic Sections
Ellipses
Applying Ellipses: Satellites Page [1 of 2]

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