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Beg Algebra: Formulas

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About this Lesson

  • Type: Video Tutorial
  • Length: 9:02
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 97 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Linear Equation Word Problems 1 (10 lessons, $13.86)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Beginning Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/beginningalgebra. The full course covers linear equations, inequalities, polynomials, rational expressions, relations and functions, roots and radicals, quadratic equations and systems of equations.
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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Formulas
Well, so, you know, we've talked a lot about solving algebraic equations. It turns out another fundamental thing we hear a lot about in our math courses are formulas for things. So I wanted to say a quick word about formulas and then take a look at some basic formulas that we may have encountered or certainly will encounter later in our mathematical lives.
First of all, let me tell you what a formula is not. A formula is not an equation. Really, when I think of a formula, I think of cause and effect. A formula is a relationship for which if you change one side, the other side will change correspondingly. So it really is causality.
Let me immediately jump to a particular example so you can see that relationship. If you think about a circle, then you may remember the area for a circle in terms of the radius. So if (r) is the radius, which I remind you is just the distance from the very center of the circle out to the edge, that's (r), then the area of that circle is r^2 which is o (r) o (r). So the area formula is A = r^2. By the way, is that special number 3.1415 and so on and it goes on forever. So that's the real number that comes up, it comes up dealing with round, circle things. So this is a formula. It's not an equation because you can't solve it for (r) and get answer. You can't solve it for (A) and get an answer, but it does have the feature that if someone gives us (r), if we know what (r) is, then I can actually insert the (r) here, and then figure out what (A) is.
Notice, for example, a big, big (r) would give a really big area. A tiny (r) would give a tiny area. That's why I think of a formula as being a causality relationship. Right, if you change this side, this side will change. So it's a cause and effect.
Let's take a look at an example. Suppose that I wanted to get some body art. Does it seem like I'm the kind of person that wants some body art? I think I am, but of course since I'm mathematically bent, my body art is going to be the most symmetric thing I can think of, which of course is the circle. So I want to get a tattoo of a solid circle. I don't know what color I'm going to pick. Maybe I'll pick blue or red. I don't know, but it turns out that they charge, the ink that's required, they charge by the area. So in order for me to figure out how much the tattoo is going to cost, I first have to figure out how much area there is, how much ink the tattoo artist will need for my body art. Well, let's suppose I want my diameter--the diameter is the length across from one side to the other passing through the center--to be 3 inches. So this is much larger. It would be more dainty on my body. It would be about like this. That looks like a nice tattoo. All right, so let's figure out what the area is. Well, the area would be easy to figure out because if the diameter is 3 inches, then I can figure out what the radius is since that would be half of it. So the radius would be 3/2 inches. So if I want to figure out the area, I just plug in (r) = 3/2, and so I see A = (3/2)^2, which I just remind you is (3/2)(3/2), and what does that equal? Well, that's times--3 times 3 is 9, over 4. So the answer is (9/4). Sometimes you write the numbery stuff first and later. It doesn't matter though. You can actually plug into a calculator using our recipe for and figure out what that roughly is in decimal form if you wanted to. In fact, let's just do that for fun just to remind you how these decimal things go. So I have a calculator here live. I'll do it for you right now, and I'm just going to take 9 and divide it by 4, and that equals something. That equals 2.25, and I'm going to multiply it by . Well, I can't actually type in because it goes on forever, so I'll do an approximation--3.14159, and I see that the area works out to be 7.068 stuff. So roughly it's around 7.
Now what are the units though? Well, since here I have the radius in terms of inches--well, here I've got inches multiplied by inches since I squared. So the units here are actually inches^2. Area actually has two degrees of freedom. So you have this stuff which is in inches in this case, and this stuff is in inches, so the area is always, in units, inches^2. So there you can see by inserting into the formula we can actually produce an answer.
Let's take a look at another example really fast. The other example is my lunch processed food has a first name and the first name actually is it's radius, and my lunch meat has a second name, which is it's height. So my question though is what is the volume of this? Well, it's actually a cylinder, right? It's a cylinder of meat product. Exciting and tasty too. How can you find the formula for the volume of the cylinder? Well, you could memorize the formula, or you could just think about the area of a disc and notice that, in fact, my cylinder of meat is really just a bunch of slices of discs, and the area of each slice, we actually know, is r^2. So if we know the area of this, all we have to do is multiply by the height, and so, in fact, what we see is we've actually generated the formula for the volume of a cylinder, and I'll write that here. So if I have a cylinder, now I'm going to do an artist's rendition of a cylinder, if the radius is (r), and it has height (h), then the volume is equal to--well, I first find the area. That's just one slice, which we already know is r^2, and then I just multiply it by how high it is. So I multiply by (h). In this case, if you measure my bologna tower, you would see that this has a radius of 2 inches. So here the radius is 2 inches. So the radius, in this tower, is 2 inches, and the height, if you measure the height, is 1 inches, which, if you write that not as a mixed fraction but as a regular fraction, we see that it's actually 5/4 inches in height. So what's the volume? Well, the volume we now calculate using the formula. Not an equation, but a formula. I have r^2, so that's now 2^2 which is 4, times the height, which is 5/4, and you notice the 4's cancel, and I'm left with just 5.
5--but what are the units? Let's see how many inches we've multiplied together. We have an r^2, so that's (r) o (r), that's inches times inches, and heights and inches. That's inches times inches times inches. This is inches^3, and volume, of course, has not only the length this way and length this way, but also it has this height. So, in fact, the units should be cubed.
Anyway, you can now find the volume for all sorts of--it doesn't have to be bologna by the way. If you want to find the volume for salami, the same formula will hold as long as it's a little cylinder, and you can see that these formulas are not equations, but if you plug in the unknowns, we can produce the right answer. Good eating to you.
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