In algebra, okay, you've heard of area or surface area, okay? Let's say you have something like a square, okay? Let's say it's 2 centimeters by 2 centimeters, okay? Now, what is the area of the square, guys? 4 what?
No. 4 cm squared. Cubed is 3. Okay? So look, in a way, not only do you multiply 2 times 2, but in a way, you multiply cm times cm. And in algebra, if you multiply x times x, right, you get what?
x squared, right? Or if you do y... Cube times y to the fifth, what would that be?
Y8. You add the exponents. It's kind of like that with units, too.
So when we're talking about area, okay, we're talking about cm or a distant square, some meter squared. Okay? Now, when it comes to volume, volume is 3d, not 2d.
So area, right, this area. is 2D. Volume is 3D space.
Okay? So let me rewrite that in pink because I'm going to kind of show what I'm talking about. Right there, there's a square. That's 2D.
So let's make it 3D. We add the extra dimension. Anybody know what axis that is called? The axis. So we have, if you look at the whiteboard, a coordinate plane has how many axes usually?
Two. What's the one that goes vertical? Y. Y, and then you have X.
If it's a 3D space, you have the one that kind of passes, looks like diagonal, but it's not quite. It's 3D. What axis is that? It's usually called Z. Okay, that's 3D.
You are 3D. You're not 2D. Okay. So now we add a length there.
So we add cm. So instead of getting the area to get the volume, instead of multiplying length times width, you multiply length times width times height to get the volume. At this point, if you are lost, you need to ask me something here, okay?
What's up? Okay, so length times width equals what? Length times width is just area for things like squares or other, um, I think they call them... I think rectangle is the generic term.
But length times width times height doesn't give you area, it gives you volume. And just like area, you do 2 times 2, except you multiply the additional 2, which equals 16. And just like with the units, you do cm times cm times cm, you get what? cm. Cubed or to the third power. Cubed is the right term.
16 centimeter cubed is a cube that is 2 by 2 by 2. Did I do that wrong? I'm sorry, 8. I did 2 times 2 equals 4. I saw that look and I'm like, I instantly, I did that wrong. 8. Because 2 times 2 is 4 times 2 is 8. Sorry.
Yes. All right. Any questions about what's on there? Yes. I know, like, when you're multiplying, you get to the positive, but it's not like when you're adding.
Yeah, when you add, like, when you add, I was using that as an analogy. But, you know, when you add exponents or, sorry, not exponents, when you add variables, let's make Y's more clear, right? You do three, that's three Y. Okay, when you multiply them, you add the exponents.
Any number that doesn't have an exponent really has the exponent to the first power, so you get y to the third. That's how it is algebraically. I'm using that as an analogy here.
cm times cm, cm, you get cubed. That's why it's cubed. So the units, what does this all mean? The units for volume is going to be some distance, some length or distance. to the third power.
Okay? When you see to the third power, a distance to the third power, automatically think volume. Now, there's another unit that automatically indicates volume, and that is the notion of liter.
Okay? Milliliters is equal to centimeter cubed. Everyone box that.
You do need to know that. So, milliliter, these are the same thing. Milliliter equals centimeter cubed. Those are the same thing.
If I say there's three milliliters of X, there's three centimeter cubed of that X, whatever that is. Water, gas, solid, doesn't matter. Okay?
We usually use milliliters for liquids. Okay? Any questions about what I just explained here? So this is all...
I need to give a better pause there. Any questions? Okay. So...
This is all the conceptual stuff, but maybe what you should think, how does it relate to mass and density? So here we have the volume is the space. Okay, the volume is the space.
Okay? The mass is the stuff that can take up in the space. Take up space. So volume is how much space it takes up.
Is the amount of space we're talking about. Okay? The mass is the stuff that takes up that space.
Does that make sense? Okay? Now, density is just this relation.
Density is a specific amount of stuff, which is what? Mass. volume, which is just volume. That's what density is. Density, if you're trying to make something very dense, you're trying to pack a lot of stuff in a certain amount of space. Okay.
So when you're going on a trip, your suitcase, you want a lot of stuff in that little amount of space, right? So what's inside a packed bag, like a suitcase is very dense usually. Okay. But an empty suitcase, isn't very dense. It's just filled with air, okay?
Air is not dense, okay? Does everybody understand this part? Because then we have a third part here.
Okay, so you have to understand, and I'm going to rewrite this, but you don't have to rewrite it. I just need more space, okay? You have to understand that if the mass does not change, sorry, let's do volume.
If the volume does not change, I mean, that slash mark represents no change for me. If the mass increases, what happens to the density? It will also increase.
You need to know that. Okay? Just use the suitcase example. The more stuff you put in that space, the more dense it becomes. That's easy.
Yes? Yeah, uh-huh. Density is a specific amount of stuff or mass in a specific amount of space. Density is a specific amount of stuff, or mass, in a specific amount of space.
Volume. Okay. So, anyways. If the volume remains and the mass goes up, density goes down. Then you should know that if the volume remains and the mass goes down, then the density will also go down.
So, suitcase. If you take more stuff out, it's just less packed and less dense. Alright, you also need to know what happens if we increase volume, but we don't increase the mass. Okay, so take the suitcase. Okay, you have 10 outfits, but you don't like them.
You know, you don't want to get it very wrinkled. Okay, if it's too dense, it might ruin your clothes and your shoes, especially. You know, don't you all hate that when you pack shoes and it...
Like smashes the shoe I had to do no one knows no one talking about Okay, well there you go, that's why okay because if you put shoes in your suitcase it will especially nice shoes it will Literally flatten the shoe and it can ruin it anyways the point is that I need a bigger suitcase Okay, so if I increase the volume, but I don't change the amount of stuff It becomes what? less dense. Okay?
And of course the inverse, if volume goes down, if I decrease the amount of space and just keep the same amount of stuff, then that's a way of increasing the density. Okay? Now, If all you know, if density increases, what does this mean?
Either what? If all you know that something became more dense, what are my options for what happened? Yeah.
Okay, right. The mass increased or what? The volume decreased. Either one. Can you know from just knowing that if density increases which one it is?
I'm going to call all random numbers in a second. Who's going to save the room from random number from this poor person that I'm going to call randomly? So the question is, if all you know is that the density increased, can you know which one it is, mass or volume?
Raise your hand. Yeah. Okay.
Right here. No. Oh, good. Good.
No. Okay. The point is that it's either one.
You don't know which one. Okay. You need more information other than just say, yeah, it's more dense. Well, you could have gotten a smaller soup bag or you could have packed in more stuff. Okay.
It could be either one. Right. So that you need to make sure that you understand these relations.
Okay. So going back to your bell ring. I want you to explain this in your bell ringer. That's how you get full amount of points Okay, so I'm gonna stop That's our problem though.
Some of y'all may already know how to solve it Some of you may not but I'm going through a step-by-step Process okay to how to solve these problems and others like it the first step is Right What is given? Right? What is given?
And I would put by each variable, by variable. Okay, so here's what I mean. They give me three kilograms. What is that a unit of?
Kilogram. What did someone say? What is a kilogram of distance, time?
What is it? Weight is, yeah, in one sense, it's not exactly weight. It's close to weight.
People in Europe, when you ask them their weight, they'll give kilograms. But technically, it's what? It's mass, right? So M equals, they give us a mass, 3 kilograms.
Okay? They also give us... the density. So we'll write d equals 10.5 grams per centimeter cubed. And suppose I ask, what is the volume?
Okay, that's what we want to know. You're going to include that in your given. Do they give you the volume? No, that's a question mark.
Okay, so suppose we want to know the volume. So you write V equals blank or question mark or even some variable. You can write V if you want. All right, the second thing you're going to do, okay, is write down the formula.
So in this case, it's density. You're going to write d equals mass divided by volume. Now, because it's volume, you might have memorized the other way to put volume. What is that equal to?
I think it was in your notes last class. It's going to be mass divided by density, right? Okay, so this is the one we're going to use. It's easier. Okay, any questions about that?
All right, let me go back to my pen. Okay, number three. Oh, I almost skipped a step.
Make sure all base Units share prefixes. That's to say all the units match. All units must match. Let me explain what I mean in a second.
I'm gonna let y'all write that down and I'm gonna explain what I mean. Make sure all base units share prefixes. And just so you know, this includes, it counts if none of them share a prefix, that's good. As long as they all share the same prefix or none of them have prefixes, it's fine. Okay, so let me explain what I mean.
Now, I'm gonna do the problem wrong. Okay, to show you why this is important, okay? Suppose, do not write this down.
I'm writing, I'm doing it wrong, okay? Okay, Mr. Felty, I'm going to use this formula, okay? I'm going to get my, I almost said velocity. Volume equals, okay, mass 3 divided by 10.5.
I just plug it in, right? And when I get that. I get about one, let me see, about, you know, 0.3 something, something, something, okay? Now, the problem with doing three divided by 10.5 is that three isn't kilograms. But this is grams.
They don't match. They don't have the same prefix. So what do I have to do? I have to get rid of kilo or I have to change grams to kilograms. It's way easier perhaps to look at this and just change that directly to grams.
So I'm going to do that third step. Okay, I'm going to change three kilograms to grams. That's going to be 3,000 grams. Three kilograms, like if I had $3,000, that's $3,000.
3K kilograms, 3,000 grams. Yeah. Did you have a question or are you good?
Okay. Any questions about that? So now, do my base units match? Yeah.
Okay, there's only one of that kind of unit, so we don't need to worry about it. But my base units match. They match the prefix.
They both have the same prefix, which is no prefix. Okay, so we did that. For, let me get rid of my colors consistent, plug in givens.
If, yeah, well, just plug in givens into formula. Okay. And so whatever formula you're using, you just plug in. Now, sometimes you might have too much missing information, and then you need to think about other formulas to get the missing information.
But you might always not be using one formula. But in our density problems, this is the main formula. So we do volume equals, what was it, 3,000?
Yeah, 3,000. grams divided by 10.5 grams per centimeter cubed Okay now when I do that the grams will cancel okay and I mean, the grams will end up canceling because you're dividing a fraction, but it'll end up canceling. You'll get centimeter cubed.
And then 3,000 divided by 10.5 is going to be about what? Let's see. Exactly. 285.7 centimeter cubed. That's going to be our answer.
So I guess the score is plug in and solve. Any questions about these kind of steps? Especially the tricky one is number three.
The third one is the trickiest because you might feel like you're doing everything right and not realize that the prefixes don't match. Okay? With that, I'm going to stop recording.
Number three, it says to share. Make sure all, no, no, what was it? All of the same base unit units need to share prefixes. Either need to share prefixes.
or have no prefix. Okay, guys, one second. I'm going to be done in a second.
I'm waiting. So, let me... Are you done?
All of the same base, you just need to share the same prefix or have no prefix. So, if I have three kilograms and then like five decigrams of something else, So either I got to convert all of these to grams or what? They each can be both. You can change it both to one to kilogram or the other to desigram. You can change this to desigram and then then they now match and that's good.
Or you could have changed it to all kilograms. OK, the point is that they all need to have the same prefix or no prefix at all. OK, because then you can't.
Look, watch. I use the example of money, right? If I have 5k dollars, okay, and then I add 5 dollars to this, right? I can't say that I have 10k. You see the problem there?
If I say, I have 5K, let me give you your $5. Now I have 10K. No, can't do that. Just like if I had 10 kilograms of something, I can't just add grams to that and say I have 10 kilograms. Okay?
Does everybody understand? Yes. Yeah.
What if they're like different? Okay, then they don't have this. They're not the same base unit. So for volume, right, you might use centimeter cubed. And then for mass, you might have kilograms.
But look, they're not the same base unit. No worries then. You don't have to worry about it. If they have the same base unit, so if you on one hand had centimeter cubed and then you had... Kilometer cubed?
There's a problem. You need to change, right? Either change both of them to what? Either centimeter cubed, change both of them to kilometer cubed, or meter cubed. These you have to change.
So, does that answer your question? Does that help? You don't change it?
Over here, you know why you don't change it, because they don't have the same base. This is volume, that's mass. So, you're not... But the fact that both of these are what?
These are both volumes, but they don't share the same prefix. You got to change it. Okay, that's it. Thank you all.