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Lecture Comments (2)

1 answer

Last reply by: Professor Selhorst-Jones
Tue Nov 19, 2013 5:56 PM

Post by Aleksander Rinaldo on November 12, 2013

Is the fixed amount of gas K?

Variation Direct and Inverse

  • Variation is a way of talking about how different things relate to each other. Depending on the type of variation, we'll know what form the relationship takes. However, we should note that variation isn't connected very closely to functions. While you can describe it in the language of functions, it's easier to talk about variation with equations, so that's what we'll do.
  • Direct variation says that two things are directly related to each other. If one goes up, the other goes up. If one goes down, the other goes down. They (usually) go at different rates, but the same "direction".
  • There are many different ways to say two things (let's call them x and y) are in direct variation:
    • x and y vary directly;
    • y varies directly as x;
    • x and y are directly proportional;
    • y is directly proportional to x.
    In any case, all these phrases mean the same thing mathematically:
    y = k ·x,
    where k is a constant. It is called the proportionality constant or the constant of variation. It is the rate at which the two things are connected.
  • Inverse variation is the opposite of direct variation. It says that two things are inversely related to each other. If one goes up, the other goes down. If one goes down, the other goes up. They (usually) go at different rates, but opposite "directions". [Caution: inverse variation is not related to inverse functions. It's based on the idea of multiplicative inverses, like 3 and [1/3].]
  • Like direct variation, there are many alternative ways to say that x and y are in inverse variation:
    • x and y vary inversely;
    • y varies inversely as x;
    • x and y are inversely proportional;
    • y is inversely proportional to x.
    Not only that, but the relationship is sometimes called a reciprocal proportion or (confusingly) indirect variation. Still, they all mean the same thing mathematically:
    y = k

    x
    ,
    where k is a constant. It serves the same purpose as it does for direct variation: it gives the rate at which the things are connected.
  • Joint variation is a variation where multiple direct variations are happening at the same time. We could say:
    • z varies jointly as x and y;
    • z is jointly proportional to x and y.
    Both of these would mean
    z = k·x ·y,
    where k is once again a proportionality constant.
  • We can also combine direct and inverse variation if we have these sorts of relationships going on simultaneously. Pay attention to what kinds of variation each variable gives, then put them all together. Notice that no matter how many variations are put together, you only need a single constant k.

Variation Direct and Inverse

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:06
  • Direct Variation 1:14
    • Same Direction
    • Common Example: Groceries
    • Different Ways to Say that Two Things Vary Directly
    • Basic Equation for Direct Variation
  • Inverse Variation 3:40
    • Opposite Direction
    • Common Example: Gravity
    • Different Ways to Say that Two Things Vary Indirectly
    • Basic Equation for Indirect Variation
  • Joint Variation 7:27
    • Equation for Joint Variation
    • Explanation of the Constant
  • Combined Variation 9:35
    • Gas Law as a Combination
    • Single Constant
  • Example 1 10:49
  • Example 2 13:34
  • Example 3 15:39
  • Example 4 19:48

Transcription: Variation Direct and Inverse

Hi--welcome back to Educator.com.0000

Today, we are going to talk about variation--both direct variation and inverse variation.0002

Variation is a way of talking about how different things relate to each other.0007

Depending on the type of the variation, we will know what form the relationship takes.0011

However, we should point out that variation isn't connected very closely to functions.0015

While this sits in the section on functions, it is only sort of connected to functions.0019

We can describe it in the language of functions; but honestly, it is much easier to talk about it with equations.0024

It is much easier to talk about variation using equations; so that is what we are going to end up using, that y =...stuff involving x,0029

what we are used to from doing lots of algebra for it.0035

That is what we are going to see for our variations.0038

In any case, variation comes up a lot in a wide variety of real-world situations; so it is important to understand.0040

If you are interested in physics, chemistry, economics, astronomy, scientific fields in general...you are going to probably need to be familiar with variation.0046

It gets tossed around in those fields; also, it tends to show up on standardized tests a lot--things like the SAT, the ACT, the GRE...0054

All of these have a tendency to have one or two questions about variation, either direct or inverse variation.0061

So, it is important to know this thing a little if you are going to be taking one of these standardized tests at some point.0068

All right, let's take a look at it: direct variation: this is the simplest form of variation, but it is also the most common.0073

Direct variation says that two things are directly related to each other: if one goes up, the other goes up.0079

If the red one goes up, we know that the blue one will also go up.0085

Similarly, if the other goes down, if the red one goes down, then we know that the blue one also has to go down.0090

They will go at different rates; for example, in this picture we have here, the blue one always goes at a smaller rate,0097

and the red one goes at a bigger rate--it goes faster than the blue one.0103

But they always go in the same direction; so direct variation is the same direction--one goes up; the other goes up; one goes down; the other goes down.0106

They are linked; they move in lockstep.0114

We see this kind of relationship in a lot of situations; here is a common, everyday example.0117

Say it costs you two dollars to buy a loaf of bread; if you buy 10, 10 would cost $20; and if you bought only 2 loaves, it costs $4.0122

The more bread you buy, the more cost; the less bread, the less cost.0131

The total cost of the bread and the number of loaves you buy are in direct variation.0135

It seems kind of obvious; but direct variation is actually a pretty simple concept to get around.0139

It is just two things that are linked: one goes up; the other one goes up; one goes down; the other goes down--direct variation.0143

There are lots of different ways to say that two things (let's call them x and y) are in direct variation.0149

We could say "x and y are in direct variation"; we could say "x and y vary directly,"0155

"y varies directly as x," which is to say "as x would do something," but we don't know what x is doing yet until later on;0160

so it is "y varies directly as x"; "x and y are directly proportional," "y is directly proportional to x."0167

There are lots of different ways to say it; but they all mean the exact same thing mathematically.0174

In mathematics, it means y = k times x, where k is a constant.0178

k is just a constant; it is called the proportionality constant or the constant of variation.0185

It is the rate at which the two things are connected.0190

Remember, with the arrows, the red arrow is bigger than the blue arrow;0193

there is some rate that says the red arrow grows faster than the blue arrow, whether it is going up or going down.0196

In the previous example about loaves of bread, k would be representing the price of one loaf.0202

That is the rate of connection between loaves of bread and the cost of the bread.0207

It is the price; price is what is connecting them--that would be our rate k for that loaves of bread example--it would be based on the price.0211

All right, inverse variation: the idea of inverse variation is the opposite of direct variation--not a big surprise there.0220

It says that two things are inversely related to each other; this means that if one goes up, the other one goes down.0228

But if one goes down, the other one has to go up.0237

They are usually going to go at different rates; once again, they have different growths in the red and blue arrow.0244

But they are going in opposite directions; that is the really key idea to get across from inverse variation.0249

It is that in inverse variation, they go in opposite directions: if one grows, the other one shrinks; if one shrinks, the other one grows.0255

And also, I want to warn you here: inverse variation is not related to inverse functions--it is not connected to inverse functions.0265

It is based on the idea of multiplicative inverses, those reciprocals, like 3 and 1/3.0273

3 and 1/3 are multiplicative inverses, because they cancel each other out; so that is what we are talking about.0279

We are not talking about functions canceling each other out--that kind of inverse function;0284

we are talking about the multiplicative inverse; that is where inverse variation is getting its term from.0288

All right, this idea comes up a lot less in everyday situations; we aren't going to see it with bread.0294

But it is pretty common in the sciences--for example, consider gravity.0299

Gravity is an inverse variation effect with distance.0303

The force of gravity that Earth exerts is inversely related to the distance from Earth.0307

The more distance you have from the earth, the less gravity you experience.0312

The closer you are to Earth--the less distance you have--the more gravity you experience.0316

Gravity and distance are in inverse variation; one of them goes up; the other one goes down; the first one goes down--the other one must go up.0321

And technically, just so we have this on the books officially, it is not actually distance that is the inverse; it is the distance squared; but it is the same idea.0329

One is going up; the other one is going down; the first one goes down; the other one goes up.0338

Technically, it is d2, but you can learn about that in a physics class; don't worry about that right now.0344

Inverse variation--there are lots of different ways to call it out, just like direct variation has many names.0349

We can say "x and y vary inversely," "y varies inversely as x," "x and y are inversely proportional,"0354

"y is inversely proportional to x," or "x and y are in inverse variation."0361

And we also will sometimes call the relationship reciprocal proportion (remember, because we are talking about flips, the 3 to 1/3 kind of reciprocal).0366

Or confusingly, sometimes, once in a while, you might hear somebody call it indirect variation.0375

This is kind of weird, because direct/indirect...I think it is kind of weird; I think "inverse" and "reciprocal" really get the idea across much better than "indirect."0380

But once in a while, we will hear people use that; so it is important that we are aware of it.0389

In any case, even if it has all of these different names, they all mean the exact same thing mathematically.0394

y = k/x; and once again, k is a constant--it serves the exact same purpose as it did for direct variation.0398

It gives the rate at which the two things are connected.0406

How much is that blue arrow going to grow or go down, depending on how much the red arrow is changing?0410

The red arrow and blue arrow, on the last one, where we talked about inverse variation--they didn't go at the same speed; they went at different speeds.0416

The red one grew a lot, and the blue one went down a little bit.0424

The red one went down a lot, and the blue one went up a little bit; so we have that same rate thing going on with this k.0428

And if we wanted to, we could also see this as y = k times 1/x.0434

That would be perfectly reasonable, as well; but I prefer k/x, just because it seems a little more compact.0440

But they are really just equivalent statements.0444

OK, joint variation: if we want, we could have multiple direct variations going on at the same time.0447

If we want to do this, we use the term joint variation; so we could say "z varies jointly as x and y,"0454

which says that z has a direct variation with x and a direct variation with y.0461

Or, we could also say that z is jointly proportional to x and y; these are our two ways of saying it.0466

But they are both going to mean the same thing: z = k times x times y.0472

And once again, k is a proportionality constant; it is the thing that is linking the rates of change between various things.0476

Now, really quickly: you might wonder, "Wait a second; if z is connected to x by direct variation, then that would be z = kx."0483

But then, we also have direct variation with y, so that is z = ky; so let's use a different letter; let's call them k1 and k2.0493

So, k1 and k2 are both constants; we are going to need different constants, because x and y are different things.0506

If we put these together, we would really get z = k1 times k2 times xy.0511

We can put those constants together, k1 and k2; so why is it that is shows up as just k, as opposed to 2 constants?0522

Well, remember: if k1 and k2 are both constants, then that means that,0529

when we multiply them together, k1 times k2 is constant, as well.0537

So, if k1 times k2 is just a constant, then that means...0544

let's just give it a new name and say k1 times k2 equals just k.0549

And that is why we only have to see k there--because it is two constants combined into one constant.0553

So, we aren't going to have to worry about keeping track of two separate constants.0558

We can just merge it into a single proportionality constant; and that is why we only have the one of them.0561

So, we have direct variation; we have joint variation (multiple direct variations going on at once)--0566

it is all of the things we are varying with one proportionality constant, k; great.0570

We can also combine multiple types of variation; we can stack direct and inverse variation if we have these sorts of relationships going on simultaneously.0576

For example, here is a gas law from chemistry and from physics; you can probably learn it in both courses, depending on how the course is taught.0584

Given a fixed amount of gas (which is to say, just air, like the kind of air in a room, not "gas" like gasoline/petroleum)--0590

given a fixed amount of air or gas in a container, the pressure, P, of the gas0597

varies directly as the temperature, T, and inversely as the volume, V.0602

So, the inverse thing would show up down here by the "divide by V"; and our direct would be just T.0609

So, it is k times T for the direct, and then k/V for the indirect; and once again, we put them together;0615

and they just combine, and they become a single constant of variation.0625

So, we just get a single constant, k; so we just put the various kinds of variation that we have...we just stack them all together.0631

We do them all at once, and we put only one single constant, k; and that will be enough, for the exact same reasons we talked about on the previous slide.0637

All right, that is everything; we have everything we need to get to the examples.0646

a and b are in direct variation; if a and b are in direct variation, we know that a = kb.0649

Or, we could also write it as b = ka; but notice that they will end up having the same effect.0657

When a equals 13, b equals 5; so let's plug this in, and let's find out what k is.0662

If we want to know what b is when a equals 52, we are going to need to know what k is, to be able to figure that out.0668

We start by figuring out what k is: 13 = k(5); so that means that 13/5 = k.0673

Great; if that is the case, we can go back, and we have the a = 13/5k; now we plug in a = 52, and we figure it out for a different b.0682

If a is 52, then we have 13/5 times k; we multiply by 5 divided by 13 to cancel out that fraction on the right side; we get (5/13)52 =...0692

oh, oops, sorry; not k; I made a mistake; a = 13/5 times b; my apologies.0705

So, we plug in...now we are trying to figure out what b is; 5/13 times 52...13 actually goes directly into 52;0713

52 can be broken up into 13 times 4, so we can come along and cancel this 13, and we just 5 times 4, which is 20.0722

So, 20 is what b is when a is 52; great.0732

In the next one, x and y are in inverse variation; if they are in inverse variation, we have y = k/x.0737

We could also do it as x = k/y; it will have the same effect--we end up having different k's,0745

but the important part is that we have this setup of the inverse like this.0750

When x = 72, y = 3; once again, we want to figure out what the k is, because the next step is to figure out what x is when y is 24.0753

So, we will need that information about what k is.0762

So, x we put in as 72; y we put in as 3; so 3 = k/72.0764

Multiply both sides by 72; it cancels out the denominator on the right side; 72 times 3 equals k.0772

We use a calculator, and we get 216 = k; you probably don't even need a calculator for that one--you might be able to do that one in your head.0778

y =...replace the k now; y = 216/x, so now we put in y = 24; 24 = 216/x.0786

So, we want to figure out what x is; we multiply the x over on the left side, and we divide by the 24; we get 216/24.0797

You probably want to plug that one into a calculator; and we will get x = 9.0804

And there we are--our answers; great.0812

The next example: p varies jointly as m and n; what does that mean?0814

"Jointly as m and n" means that these two are going to be on the right; so p = k (we have to have that proportionality constant), times m, times n.0819

When m equals 4 and n equals 8, p equals 2; then we want to find out what n is when p equals 60 and m equals 24.0829

So, once again, we need to figure out what that k is, what that proportionality constant is,0836

if we are going to be able to figure out what n is in the second half of the question.0840

So, we plug in all the values that we know from the beginning; we have p = 2; 2 = k times m at 4, and n at 8; 2 equals k times 32.0844

Divide by 32; 2/32 = k; 1/16 = k; great.0857

So, with that information, we can now take this; we will create a new thing that tells us the relationship in general.0866

p = 1/16 times m times n--great; we know that p is 60; we know that m is 24 for the one where we want to figure out what the n is.0871

So, we plug in p = 60; that equals 1/16; times...we don't know what n is, but we do know what m is: m is 24 times n.0882

Now notice: there are some common factors between 16 and 24; 24 can be broken down into 8 times 3; 16 can be broken into 8 times 2.0892

So, the 8's cancel out, and we are left with 60 = 3/2 times n.0903

We can now multiply both sides by 2/3; 2/3 times 60 = n.0912

So now, that cancels out the fraction on the right side; we can break down 60 into 3 times 20.0920

The 3's cancel out, and we have 2 times 20, which is 40, equals n.0928

There is our answer; when p is 60 and m is 24, n has to be 40.0934

The next example: all right, this one is a little more complex, because it is a word problem; but we will be able to work through it, actually.0939

Hooke's Law connects the force of a spring to its compression; it says that the distance, x, that a spring is compressed or stretched0944

from its equilibrium natural length is directly proportional to the force, F, of the spring.0953

First, let's understand what that means: the very first thing you should do, if you read a word problem,0959

and you are not sure what a word means--go look it up.0964

Either (if it is one of the words that you learned because it was in that lesson) go check what that definition meant in your math book,0967

or from this lesson; or if it is just a word that you don't know, that didn't show up in the math before, just look it up in a dictionary.0973

Equilibrium: say this is a spring--equilibrium is just what it is when it is at rest, when it is not being compressed or stretched.0979

It says that the distance, x, a spring is compressed--if we push it in by some amount x, it compresses it.0987

And then, that causes the spring to push back with some force, F.0992

And this law, Hooke's Law, says that the force, F, is equal to k times that distance.0996

It is directly proportional; so it has that proportionality constant, times the thing that it is directly proportional to (direct variation).1006

F = kx; now, once again, once we have that part figured out, it is just a normal problem from there on.1015

Say a spring is stretched by .1 meters and has a force of 95 Newtons when stretched that far.1022

And Newtons is just the unit of force in the metric system--just force.1028

And then, what force would it have if stretched by .25 meters?1032

Well, .1 meters is just x = 0.1, and force equals 95 for the first part of the problem.1035

For the second part of the problem, we have x = 0.25; and we want to find out what the force is: F = ?1042

So, we see this diagram; we push the spring in by some amount; and that will cause a force to appear,1052

in reaction to that, where the spring is pushing back; and it will depend on the amount we have compressed it.1058

If you take a spring in your hands, and you push a little bit on the spring, it pushes a little bit back.1062

If you push the spring really, really heavily, it pushes really, really hard back; and that is why we have F = kx.1066

How hard it pushes back is directly connected to how much it has been compressed (or stretched).1072

So, what force would it have if it was stretched by .25 meters?1077

To figure out that, we are going to first have to figure out what k is.1080

So, we do that by plugging in our numbers; our force is 95 when it has been stretched by 0.1.1084

We divide both sides by 0.1: 90/0.1 = k; so that just moves the decimal place over 1, and we get 950 = k.1093

So, that tells us, for this specific spring, the spring that this problem happens to be about--it has a constant of 950.1103

It has a coefficient, a proportionality coefficient/constant, of 950 (technically, 950 Newton-meters--it is just that it has units).1110

But we don't have to worry about that; 950 = k is perfectly enough.1117

So, 950 = k is what it is for this spring; however, this is true, in general, for any spring: F = kx is a very good rule1122

for describing any spring that can be compressed or stretched.1129

But this specific spring that we are working with has 950.1133

A smaller, easier-to-push-around spring would have a smaller k, and a harder, thicker, heavier,1135

harder-to-push-around spring would have a larger k.1141

So, 950 = k for this one; so now we have F for this specific spring: F = 950x.1144

So, what is the x that we have for the second half of the question? x = 0.25, so F = 950(0.25).1153

We plug that into a calculator; and 1/4 of 950 is 237.5; what are the units that we are using?1163

We were told that the unit of force in the metric system is the Newton; so it is 237.5 Newtons.1173

And there is our answer--great!1184

All right, the last example: The maximum load a horizontal beam can support, if held up at both ends,1187

is jointly proportional to the width of the beam and the square of its depth, while inversely proportional to its length.1195

Wow, let's see if we can figure out what that means before we try to do this.1202

First of all, we have some horizontal beam; we have some beam, and it is being supported at both ends.1206

It says "the maximum load a horizontal beam can support..." so there is some big, heavy weight in the middle of it.1214

And it is being supported on both ends; it is being held up just at the ends, and it is able to support an amount.1225

The maximum load is jointly proportional to the width of the beam and the square of its depth, while inversely proportional to its length.1234

What is length, width, and depth? Well, let's take another look at the beam.1244

We could have a beam like this; its depth is how far down it goes.1249

Here is d, the depth; we could also talk about what its width is; its width is going to be1260

(sorry, that was just me trying to make an arrow pointing at the w, but it ended up looking like a w, as well)1270

width...it is jointly proportional to the width; and then finally, the length of it is there.1276

Now, this makes sense; it would make sense that it is going to be inversely proportional.1287

The farther out the beam gets...we have a beam like this that is the exact same thickness and depth and everything.1293

And we support it at the two ends; it makes more sense that it is going to be easier to snap it in the middle.1301

The farther and farther we stretch it out on the sides, the easier it is going to be to snap it in the middle.1304

On the other hand, if it is all the same thing, and it is a very short thing, and it is very stout,1309

it is going to be able to support a lot more load, if it is supported very, very closely, like this.1313

So, that makes sense; it also makes sense that, if it is wider (there is more stuff there), then it is going to be able to support more.1319

And its depth...it is going to be able to support more.1328

Now, it turns out that it is not just the depth, but the square of the depth.1330

So, we have to integrate that into our formula; so let's figure out how we can turn this into just a formula in math.1335

The maximum load is equal to...the first thing we have to do is put in that proportionality constant, k.1341

k times...what is it? It is jointly proportional to the width of the beam and the square of its depth,1348

so times the depth squared, because it is the square of the depth; divided by (because it is inversely proportional to) the length.1356

This gives us a formula for figuring out maximum load.1363

We would have to know what k is to be able to use this formula; but we have a formula for doing that.1367

All right, now let's continue with the problem and see if we can use that to figure out the rest of these questions.1371

OK, so now that we have our maximum load formula here, if a given beam can support a maximum load of 750 kilograms,1376

how much could it support if its length is tripled, or its width is doubled, or its length is doubled, its width is halved, and its depth is tripled?1385

Wait a second; we don't have any information!1395

The way we did all of these previous problems was that they gave us enough to figure out k,1397

and then we used k to figure out the rest of these problems.1402

We don't have any specific numbers to work with; so what are we going to do?1404

Well, the first thing we need to do is not get scared; we have this problem, so there is a good chance that we can solve it.1407

So, let's just try the things that we normally try with word problems.1412

Let's try to just name things that we aren't sure of, at least.1415

We don't have a specific number for the length that it is; a given beam must have a length; it must have a width; and it must have a depth, at first.1417

We don't know what they are, but we can still give them names.1425

This is a great thing to do--to just give names to the things that you don't know.1427

So, let's say, right from the beginning, that we will name its initial width wi, its initial depth di, and its initial length li.1431

These all end up being the initial width, depth, and length, respectively.1444

Now, what do we know when we use the initial width, depth, and length, respectively?1457

We know that 750, the maximum load, is equal to k (we don't know what k is), times wi, times di2, over li.1460

Well, wait; I still can't figure out what k is.1473

So, what are we going to do? Once again, don't get scared yet.1475

Let's actually try some of these out: the first one is if its length is tripled.1478

So, if its length has tripled, we are going to have a different thing than li; but it is going to be connected to li.1483

If we triple the length of li, it is going to be 3li, so it is going to be 3 times li.1490

What is it going to be if we have k times di times wi, over li(3li)?1496

Well, we don't know what that is; but oh, that looks a lot like this; that is just 1/3--we can pull that 1/3 out.1508

We have 1/3 times (k times wi times di)2, over li.1516

We already know what this is; that is just 750, so it is 1/3 times 750; 1/3 times 750 is 250.1526

What is the unit we are working with? We are working with kilograms as our unit.1535

So, the maximum load, if we were to triple the length of this beam, would be 250 kilograms.1538

Great; so now we have an understanding of how this is working out.1543

We just plug it in, and we can use what we have here; we can use the information that we have.1546

We don't have to know all of the numbers; knowing just one of them is enough, because we have a general form that it is working in.1550

When the width is doubled, it would be 2 times the initial width, wi; so that would be k times 2wi times di squared, over li.1555

So, we pull that 2 out; we have 2 times (k times wi times di) squared, over li.1568

We plug in what we know that all is; that is 750, so it is 2(750), or 1500 kilograms; that would be the maximum.1578

Great; on the last one, this is a lot of things: length is doubled (that would mean 2li);1588

width is halved (which would mean 1/2wi); and depth tripled would mean 3di.1596

OK, so how does this come out? We have k times w; what w are we using? We are using 1/2wi.1607

Times d...what is our new d? It is 3di; and this is important.1618

Remember, we are not plugging in for just 3di squared; we are plugging in all of this--it is what all of the new depth is.1623

And the new depth is 3di, so it goes in in parentheses: (3di)2, so that 3 is going to get squared, as well.1629

Divided by 2li: so k times 1/2...let's pull the 1/2 down; so we will get wi up here,1637

but it will be divided by 4li times 9di2.1645

So, we can pull out the coefficients, and we will get 9/4k times wi times di2, over li.1650

We know what that one is; that one is 750; so 9/4 times 750...1659

plug that into a calculator, and we get 1,687.5 kilograms; and there we are.1664

One thing I would like to point out: notice that depth, by far, matters the most, because it is depth squared.1675

So, you can get a lot more load by just having a larger depth.1682

You have to increase width a lot more than increasing depth, because depth gets squared.1685

This is why, if you have ever worked in roofing, or anything where you see a beam supporting a long horizontal length,1690

if you get the chance to look up inside of an attic or inside of a roof,1695

at what is actually holding up the structures, you will notice that the beams aren't flat like this.1697

They are supported like this, so that they can have the most depth, because it is the depth squared that makes the strength.1702

So, they are always supported on a long, deep kind of axis, because that gives them the most strength.1708

So, if you have done construction, you have actually seen this before.1714

You have seen something where you realize that what you are seeing there is mathematics in effect in the real world--pretty cool.1717

All right, I hope variation makes sense; and we will get started on the next section in the next lesson.1723

All right, see you at Educator.com later--goodbye!1727