Grade 12 Math Live Class | Members Only

追加: 2026-05-13

[00:01:19]All right. Good afternoon. Good evening.

[00:01:29]It's not me anymore.

[00:01:32]Valerie, what what what is it in Africans? How do you say is it huh or the other valer? The original valer, which one is it?

[00:01:54]Who's good at Africans, guys?

[00:01:58]Uh to our YouTube audience, thank you for joining us. uh you we are live tonight and as always you know we deliver when it comes to content. Today we are going to be touching on optimization and this is what we uh call the application of calculus. So what we are going to be doing is um I've already introduced it in my class. Um so today what I just want to do is to just delve into some questions that I thought were interesting and you guys uh do you have questions that you want to ask? Good evening Z.

[00:02:38]>> Good evening sir.

[00:02:43]Much better today. Ah, you were going you know the other day.

[00:02:51]Hashem >> I'm much better.

[00:02:55]>> Much better. Okay, great.

[00:02:58]>> Okay. Um, if you guys have no other questions, I'd love to start with the first question and I really want us to today kind of look at different relationships that we have. Um the questions that I have prepared for you is they are just on another zinger.

[00:03:19]Okay. Right. So today I want us to focus a little bit on speed, distance and time questions. Now but of course we are going to deal with shapes as well um uh you know with perimeter.

[00:03:35]Now I want you guys to please remember the following that if I'm given a formula let's say they say to me this is the distance or the displacement of something right so if we let's say say the distance is 3t ^ 2 + 4t minus 5 okay just making an example with that okay now remember what is velocity in relation to speed so if you Remember, for those of you who do physics, we know that velocity is the rate of change in displacement.

[00:04:15]So, if I change displacement divided by time, right? That is going to give me my velocity. But remember, if I drew this as a graph, what is the rate of change? That's the gradient. So meaning that if I take the gradient of um sorry of displacement or distance right so if I say this is 60 + 4. Now remember that the display uh rather the rate of change that is the gradient in this case gives us the velocity.

[00:05:01]So anytime that they ask for the velocity, it means we're going to take the gradient. Okay? And then if I take the double derivative of speed. So the gradient of speed, right, is now going to give us the acceleration.

[00:05:22]So when I take the double derivative, that now gives me the acceleration of something. All right? Now it's important for us to know this so that we know how to apply it and to apply it correctly.

[00:05:37]All right. So remember we said the first derivative gives us velocity.

[00:05:42]Okay. And we know that the second derivative gives us acceleration.

[00:05:49]Right? This is important for us to note when it comes to examples that have to do with displacement uh or in this case with velocity and time. Right? So we've got one such an example here. I wanted us to take that they say we've got a particle that moves along a straight line. Okay. The distance in meters. So there it is. They're giving us the distance of the particles from a fixed point, okay, on the line of the uh of of time t seconds where t is greater than zero.

[00:06:27]And I mean, they don't need to tell us that twice because what is time that is less than zero? We never have negative time unless you guys can go back in time. Wish you, Philillip, don't you don't you wish you could go back in time?

[00:06:46]>> Hey, I do sir.

[00:06:49]>> What would you fix if you had to go back in time? Would you be able to tell us?

[00:06:54]>> Maybe maybe not on YouTube. Maybe not on YouTube.

[00:06:57]>> Not not here. Only in private.

[00:07:01]Okay. Uh, Lutano, good evening. What would you fix if you could go back in time?

[00:07:13]>> You'd fix a lot.

[00:07:16]>> Yeah. A lot.

[00:07:19]>> This is serious during the people. All right. So, they say to us, we've got an equation which is s of t which is 2 t ^2 - 18 t + 45. Now what did they give us?

[00:07:35]They gave us the equation for the displacement or for the distance.

[00:07:41]So they say to us calculate the particles initial velocity.

[00:07:48]Ah may I ask if I say initial velocity at which time is this?

[00:08:00]>> At 0 seconds.

[00:08:01]>> Yeah. So it's 0 seconds right but we are looking for the velocity go what did we how did we say we get velocity when we are given distance >> it's distance over change in no it's change in x over change in t >> yes change in in s over change in t it means we are taking the derivative where time is equal to zero so let's find out the derivative first. So this is going to be uh applying our derivative rule or our um so that's going to be 2 * 2 that is going to be 4t - 18. Okay, but we want the derivative when time is zero. Okay, so that's going to be 4 * 0 - 18 m. Okay. So it means in this case what was the particles initial velocity?

[00:09:08]Makes no sense.

[00:09:10]Okay. It was8.

[00:09:14]Huh? I'm not too sure about that negative. Exactly. What what does it mean? Uh but our initial oh initial velocity sorry velocity is in meters/s right?

[00:09:32]Yeah. So our distance is in meters and our time is in seconds. So which means that our h our velocity at zero time is8.

[00:09:45]Oh yes I was thinking about distance. So which means if we are assuming that here's a straight line and this particle is moving let's assume that the original distance I mean dis um rather uh velocity is to the right. So what does that mean? It means initially it must have been moving in the opposite direction. Okay. Right. For those of us who know physics, you know that uh when we say it's a negative velocity, it means that we must have been moving in the opposite direction. Okay. So, they didn't give us reference here in terms of uh velocity. So, I'm going to leave it as just negative. All right. Are we all good in the hood?

[00:10:34]Serite.

[00:10:37]Okay. Right. Valer, right?

[00:10:43]>> Yeah. So, I'm like trying to understand a bit, but I think I'm getting it.

[00:10:47]>> Okay. Tell me where did I lose you.

[00:10:50]>> No, like you didn't lose me, but I'm just trying to like make sense of everything.

[00:10:54]>> Make sense of it. Okay, that's fine.

[00:10:56]Now, they say to us, determine the rate at which the velocity of the particle changes at t seconds. Now notice what are we ch what is the rate of change now the rate of change in velocity. Now remember velocity is the rate of change in displacement.

[00:11:24]But what is the rate of change in velocity? That is acceleration.

[00:11:29]Right? So that is change or or rather the rate of change in velocity. So they said to us uh they want us to find out for 10.2 the rate of change of velo at which velocity of the particle changes at t seconds. So how do we find the derivative I mean the the the rate of change of velocity which is the acceleration that's the double derivative. Okay the second derivative.

[00:12:00]So this is going to be right the derivative of the derivative we know that this is going to be 40 minus 18. So which means our rate of change of velocity is simply going to be four.

[00:12:20]Are we all together ladies and gents?

[00:12:25]Sister right.

[00:12:28]Okay. So please repeat how you got too full.

[00:12:32]>> Hey nana.

[00:12:35]>> So remember so here I think the biggest thing is language right. So we need to understand let me I I just want to write that down quickly.

[00:12:50]So we need to understand in terms of our language that we've got first of all distance or displacement which is the original equation equation right?

[00:13:14]But we know velocity.

[00:13:18]We said velocity is the derivative. So we take we said for us to get velocity, it's always the first derivative. But remember when we use English to define what velocity is. What is velocity? It's the rate of change of displacement or in displacement.

[00:13:45]Okay.

[00:13:48]Displacement or distance.

[00:13:53]Okay.

[00:13:55]Well, displacement um yeah, let's say displacement. Okay.

[00:14:00]And acceleration.

[00:14:04]How do we get acceleration?

[00:14:07]We said this is where we are going to take the second derivative. But in English what is uh acceleration?

[00:14:15]Acceleration is the rate of change in velocity.

[00:14:25]Okay. So the one is defined by the other. Distance on its own. Velocity is the change in distance over change in time. Acceleration is the change in velocity over change in time. Now in our question Z they said determine the rate at which the velocity changes. Does that make sense?

[00:14:49]So once we say rate of change in velocity, it means we are talking about acceleration and we said how do we get the acceleration uh of of uh the particle?

[00:15:04]This is where we take the second derivative. So I'm going to take this formula here, right? And find the derivative of it.

[00:15:15]Remember a constant disappears and then 4t just simply becomes four in that case.

[00:15:24]Are we all good?

[00:15:27]>> Yes.

[00:15:28]>> All right. Not a problem. Right. Now they say to us uh in the next one they say after how many seconds will the particle be closest to the fixed point? Okay. Now remember what I said to you about uh optimizing right we said optimization you are either minimizing or maximizing but in this case we want to know closest to the fixed point right so now what do we do to find h the time at which it will be closest so to optimize the distance so we are minimizing the we take the derivative and we equate it to zero. Okay, so we know in this case we're going to say that's 40 - 18 and this is equal to zero. So 40 is equal to 18. And if we divide both sides by four, that's going to be t is uh 9 / 2, which is 4.5 seconds.

[00:16:48]Are we all together, guys?

[00:16:51]Right. Right. So, this is just what I want you to remember. When it comes to distance um we always the first derivative is velocity and when we come to the second derivative that is where we're going to say that is the acceleration.

[00:17:08]All right. Now let's go to 2025 the November exam from 2025. It has a very interesting question on volume. Right.

[00:17:23]But these are your favorites. I know on Saturday you loved them. Okay, let's go.

[00:17:30]They say we've got a rectangular metal sheet that has a dimension x and h units.

[00:17:40]Okay, so this is the guy.

[00:17:44]They say x is greater than h. So which means it's rectangular, right? In this case, it's not a square. So, it's a rectangle because they did tell us that x is longer than the um the height h.

[00:18:00]Okay. And they say and a perimeter of 50 units.

[00:18:07]All right.

[00:18:13]>> So, how do we get the perimeter of the sky?

[00:18:18]We need to have a a an expression for the perimeter. You agree?

[00:18:22]>> Yes, sir.

[00:18:23]>> Okay. Right. How do we get the perimeter >> for the rectangle?

[00:18:28]>> Yep.

[00:18:29]>> So, we're going to add h 2x and then we're going to say 2 h + 2x.

[00:18:35]>> All right. So, it means that we are getting the entire boundary line of this uh of our rectangle. So in that case it means that's h plus another h this side that's x plus another x that side. So you are quite right. So we've got 2 x + 2 h and they said this is equal to 50.

[00:19:07]What are we going to use this for? How is it going to help us?

[00:19:14]Okay, let's divide by two and we've got x + h is equal to 50, right? And let's make that equation y.

[00:19:27]How is it going to help us to get this parimeter?

[00:19:34]So, we're going to use like the rectangle is part of the the cylinder.

[00:19:39]It's part of the two the cylinder is made up of two circles and a rectangle.

[00:19:44]So the perimeter of the rectangle is going to help us find the part of the perimeter of the cylinder.

[00:19:49]>> Right. Beautiful. So they say the metal sheet is rolled into a cylinder with two open ends. So it is a hollow cylinder, right? So meaning that we don't have the top and the bottom part. It's just a the cylindrical part. Right? And in this case they say with two open ends top and bottom and a height of h units. And it makes sense that the the height would be eight units since this was um or rather since our rectangle had a height of h.

[00:20:26]Right?

[00:20:28]Now they say show that the volume of the cylinder is given by that expression.

[00:20:35]Right? What do you notice in the expression? in the expression we do not have a H in there. Right? So which means we did something to get rid of the H.

[00:20:49]Okay? Right? So firstly, let's make H the subject of the formula here. Why do we want to make it the subject of the formula? Because we want to make sure that we are going to eliminate one of the v variables. Yeah.

[00:21:12]Yes. Please talk to me.

[00:21:16]>> I wanted to ask if you weren't supposed to divide 50 by two. Also, >> I see it. You know, the moment you said it. I mean, the moment you raised your hand, I knew someone wants to give me a hiding.

[00:21:30]>> Right. So, so we divide everything by two. Thank you for that.

[00:21:36]At least one person is awake in this class as as the others I'm charging you.

[00:21:44]Okay. Uh so age is 25 - x.

[00:21:51]Right? So now we've got the expression for h. Right? Now you must keep in mind what we're trying to do is get to the volume of the cylinder.

[00:22:03]How do we normally get the volume of a 3D object?

[00:22:10]We always say the area of the base multiplied by the perpendicular height.

[00:22:20]All right. Our base is a circle and in our Oh, no. No. Before Before I ruin it, before I ruin it. Okay. Right.

[00:22:33]May I ask guys how what would be the radius of this circle?

[00:22:44]Um yeah, what would be the radius of our circle if >> Excuse me.

[00:22:55]>> Can I try?

[00:22:56]>> Yeah. Yeah. Yeah. Yeah. Go ahead.

[00:22:58]Isn't it half of eggs?

[00:23:03]>> Okay. All right. Uh, Philip as I don't know.

[00:23:12]>> Okay. All right. Now, do you do you agree with me that we had an expression for um remember we rolled the sky, right? We we rolled the uh um go what do we call it? The the the what the the the the rectangle. We rolled it up and we made it into a circle. Right.

[00:23:44]Um, so I don't think it will be a cir I mean I don't think it will be because think about it tando what you're saying is that you would have divided this guy up in half.

[00:24:00]Okay.

[00:24:02]But would you also agree with me that um the circumference of the >> equal to x? Sorry. So it's equal to x.

[00:24:19]>> The circumference is what's equal to x.

[00:24:21]>> Yes.

[00:24:24]So the circumerence is equal to x. Now how do I get the circumference of a circle?

[00:24:32]talk to me.

[00:24:37]>> I'm not sure sir.

[00:24:38]>> Okay. So the circumference of a circle remember is 2 pi r.

[00:24:44]Right? So let's do this. So the circumference of a circle 2 pi r.

[00:24:53]Do you agree with me goody? This will be equal to x.

[00:24:58]Correct?

[00:24:59]>> Yes sir. Ah, there we go. So now what we need to do is make an expression for r in terms of x. What will be r?

[00:25:12]r will be x over 2 pi. Do you guys agree with me?

[00:25:19]Right? So the circumference of our circle remember that circumference is 2 pi r. Right?

[00:25:26]And in this case we know the circumference will be equal to x and so the radius will be equal to um 2 pi r which will be equal to x and so r will be x / 2 pi. Now let's talk about volume. We said volume of a cylinder will be expressed by the area of our base. What's the area of a circle?

[00:25:58]This is pi r² multiplied by the perpendicular height which is h. Okay. So now that we know the value of r. So this is going to be pi.

[00:26:17]What is r? r is x over 2 pi x. Sorry.

[00:26:25]Over 2 pi squared, right?

[00:26:31]And this is multiplied by h. What is h?

[00:26:36]h is 25 minus x. Guys, are we good in the hood?

[00:26:43]Sis, right? Right.

[00:26:47]So, let's try and simplify this. This is pi into x^2 / 4<unk> 2 * 25 sorry min - x right let's try and this one pi will cancel out the other pi so I'm left with x^2 over 4<unk>i I into 25 - x. Right? I'm deliberately being slow guys so that you can you I I don't lose you. Uh so that you can see what I'm doing. Now of course we now going to use our distributive rule right to try and simplify this even further. So this is 25 over 4 25 * x^2 over 4 pi. This will give me 25 x^2 / 4<unk>i - x cubed over 4 pi. Now let's check what is that what they asked over there.

[00:28:04]It is definitely that. Does it make sense guys?

[00:28:11]I know we took a little bit of creativity nana but um yeah so it's really important that that we get this right. Okay. Right. And we know where this leads up to. They will now ask us to optimize and conj How do we optimize Valerie the other Valerie that is not the technical producer?

[00:28:49]How do you maximize?

[00:28:53]>> I may have.

[00:28:55]>> Okay. So, please remember what did we say? We said anytime we want to maximize we always take the derivative and make it equal to zero. Okay. Right. Oh yeah.

[00:29:08]>> So once once they get to it once we are able to derive an equation then the next rational step all the time you you you'll see it every time they'll say find the value of x or they'll say find the value of whatever such that you've got maximum area maximum volume maximum whatever. So in this case or minimum sometimes right like we did in the previous question. But we know if we're maximizing or minimizing, we're always going to take the derivative and make it equal to zero. So let's take the derivative and make it equal to zero.

[00:29:44]What is our derivative? Remember we multiply by the power and we subtract one. So 25 * 2 gives me 50 over 4.

[00:30:03]uh 50 over 4 pi x. Remember we said minus one and then minus so 3 * 1 will be 3 x^ 2 remember this is -1 over 4 pi and this is equal to zero. All right. So let us uh let us let's try and simplify this. Okay.

[00:30:42]Right. I realize that we've got 4 pi both at the bottom as well as at the top. Right. So is it okay if I say let's take 1 / 4 pi x.

[00:31:05]Now what are we going to be left with in the first bracket?

[00:31:09]I'm going to have 50 x uh sorry not uh not not 50 rather not 50x but rather 50 minus 3x and this is equal to zero.

[00:31:29]Right? You could have solved this in any other way uh that you prefer. Um, for instance, if you wanted to, you could have taken out sorry, you could have taken out just x as the as as the coefficient. Okay, but either way, we would get to the same answer. Right?

[00:31:48]Now, let's find the last part. So, 1 / 4 pix is equal to zero.

[00:31:58]Right? If I multiply by 4 pi, what I do on the left, I do on the right.

[00:32:06]So, x is equal to zero. Now, is it possible for us to still have a cylinder, but yet the dimensions are zero?

[00:32:17]That's not possible. Right? So, which means this is not applicable.

[00:32:24]Okay? So, this would not be applicable.

[00:32:28]Or here we've got 50 is equal to 3x and we can divide both sides by 3 and x is equal to so we've got 5 over 3 I mean uh 50 over 3 rather and that gives us 16.67 6 7 uh I think that was what in meters units. Okay. They didn't tell us uh what units. So we're just going to use units generically. Okay. Right guys, are we okay so far?

[00:33:14]All right. Awesome stuff. The original Valerie is trying to tell us there's a question on YouTube.

[00:33:24]Can I just quickly check there?

[00:33:28]Um, all right.

[00:33:34]Someone is saying, "Can I start again?"

[00:33:37]Where? Start what again? I wasn't really I'm not sure when that comment came about.

[00:33:49]Hey.

[00:33:51]Okay. um uh of of the question.

[00:34:02]Okay. All right. Okay. Awesome stuff.

[00:34:06]Okay. Right. Okay. Can I just quickly check from the guys in my class? Are you guys did did you understand?

[00:34:17]Okay. I'm getting a sharp h the other valer is not giving me a sharp val.

[00:34:24]>> So like I said before I'm trying to make sense of everything because it's my first time learning about velocity.

[00:34:31]>> I see about about optimization in general.

[00:34:37]>> Okay. All right. Okay. Um All right. I'm going to go through this again. Right.

[00:34:44]Just need to show you. All right. Please do allow me. I won't go through the entire question, but I'll actually just go walk through what I did. Right?

[00:34:58]So, we are taking a piece of paper. So, if you can imagine, right? Me being the teacher that I am, I'm always trying to find stuff that I can use. Okay? So, I'm taking this piece of paper.

[00:35:17]There it is.

[00:35:19]Taking this piece of paper and I'm going to roll it up into a cylinder.

[00:35:26]Okay. So, uh some people are thinking about stuff that is being smoked.

[00:35:35]Okay. So, roll it up into a cylinder.

[00:35:49]Tell us where you know it from.

[00:35:52]>> Don't sir. Why do you know it?

[00:35:58]>> How do I know?

[00:35:59]>> No.

[00:36:02]>> I've been alive for a very long time.

[00:36:06]I've seen things.

[00:36:09]Okay. So, you're taking this piece of uh paper and you are going to roll it up into a cylinder. Now, remember the part that was the let's say the top and the bottom part.

[00:36:27]When you roll it up, that's the part that makes the circle, right? This is the part that makes the circle. which means I know what the circumference of the circle is. Okay, so the circumference of our circle is going to be or rather the formula for the circumference of a circle is 2 pi r.

[00:36:51]Right? That's the formula for the circumference of a circle. But this 2 pi r is equal to x because x is that dimension of the uh the bottom part of the of of of our of our cylinder. Okay.

[00:37:11]Right. So in this case r so I'm making r the subject of the formula. So it means that r will be equal to x / 2 pi.

[00:37:23]Okay. So that is what I get in terms of r right and then they also told me that the perimeter is equal to 50 units. Now remember what is what is the perimeter? We are adding all the sides together. So, it's going to be x, remember the top and the bottom are the same, plus h plus another x plus another h. So, we end up with a formula that says 2 h + 2x.

[00:38:03]This will be equal to the perimeter, which is 50 units that was given to us.

[00:38:10]But when we're doing optimization, we are trying by all means to end up with only one variable. So let's try and get rid of one of them variables. So we're going to get rid of h. So we need to make h the subject of the formula.

[00:38:25]Divide everything by two here. I've got h + x which is equal to 25.

[00:38:37]Okay. Right. And so we get h is equal to 25 - x. Why am I doing that? I need to find an expression that I will substitute for h that will get rid of h.

[00:38:56]So the final part we are finding the volume because that's what they want.

[00:39:05]They said the volume of the cylinder.

[00:39:08]How do you get the volume of a of a shape? Always say the area of the base multiplied by the perpendicular height.

[00:39:18]What's the area of a circle? Right?

[00:39:22]That's pi r 2 multiplied by the height which is h. And the final part that I did was okay for r I've got r in terms of x that's x over 2 pi.

[00:39:39]So that's x / 2<unk>i all squared multiplied by h and h is 25 - x. Right? The rest guys is history because then I've got the expression that they want over there. So anytime I want to maximize or minimize, I take the derivative and make it equal to zero.

[00:40:04]Scooty nice. All right. I hope that uh for those of you that are listening in on YouTube.

[00:40:11]Okay. Right. So we are going to do that.

[00:40:15]All right. Now let's do this quickly. I want us to take one um actually. Yeah.

[00:40:24]Uh I had a question here.

[00:40:27]from the IB that looked very interesting as as well. Right. I want us to quickly have a look at it together.

[00:40:36]Right now they say to us the diagram represents an inverted right square pyramid. Okay.

[00:40:48]So they say the length of each side of the square is 3 cm. So this is a square and each of the sides is 3 cm but we're taking another side and this side is 3 cm.

[00:41:13]All right. Now we are saying in this case uh the length of each of the sides is 3 cm and the perpendicular height of the pyramid is 9 cm. Now remember I can't really draw this nicely but the perpendicular height is in the middle here and it goes all the way up until the top. Okay. So if you think of your pyramid over here, okay, I want to try and draw a the flip side of our pyramid. Okay, the top part of our pyramid. So the top part would look like this, right? And of course it's made out of uh that.

[00:42:01]Okay?

[00:42:03]Right? So the height which is the perpendicular height from the top to the center of your pyramid which is h right they said to us this is 9 cm that's 9 cm okay right I hope that makes sense they say to us a new pyramid is to be constructed so that the volume of the pyramid is maximized.

[00:42:36]Okay. Right. They say a new pyramid is constructed according to the following rules.

[00:42:45]Firstly, the pyramid must retain a right square pyramid. So meaning the new pyramid must still be a square on either side. The perpendicular height of the pyramid must be decreased. So we want to make the height smaller, right?

[00:43:08]Uh by the same amount um sorry it must be decreased by the same amount that the length of each side of the square is decreased.

[00:43:21]Hi guys.

[00:43:24]So if I am going to decrease each of the sides, I need to remember each side was 3 cm.

[00:43:32]3 cm.

[00:43:35]Let us say now we are going to decrease each side by a length of x.

[00:43:44]So which means your new pyramid please listen very carefully has a length of 3 - x.

[00:43:55]3 - x right remember it's still a square.

[00:44:01]They said the height we must h reduce the height must be decreased by the same amount of that the length of each side of the square is decreased. So which means we need to decrease the height by x. So now what is the new height? The new height will be 9 - x because the old height was 9 and we are decreasing by the same length which is uh the same length as the two sides. Okay. Now they say to us >> sir they said we must increase the side of each square.

[00:44:50]Are we >> excus?

[00:44:54]>> Okay. Sorry.

[00:44:57]Okay. Let's read that again. The perpendicular height of the pyramid must be decreased by the same same amount as the length of each side of the square is increased.

[00:45:12]This makes sense. Thank you so much, Dandu.

[00:45:16]Right. So that is 3 + x. Okay. Meaning that we are decreasing the height by x but we are making the circle bigger.

[00:45:29]Okay. We are making the circle bigger.

[00:45:31]We are making the square uh we are increasing each of the side by three by x as well. Okay. Right. And now they say to us find the ratio of the length of each of the sides right the find the ratio of the length of the side of the square y English guys find the ratio of the length of the side of the square to the perpendicular height when the volume is a maximum.

[00:46:07]Okay, the ratio meaning we are dividing right the length of the side with the perpendicular height. Simple as that.

[00:46:20]Okay, they can say it in expensive English but we've understood what they mean right now. They give us a formula for the py the volume of a pyramid.

[00:46:31]Okay, so let's go for it. So the volume in terms of x now remember this is the volume of the new pyramid this is 1 / 3 multiplied by the area. Now what is our area right? It is the area of our base multiplied by the height. But what is our base? Our base is a square. So this is going to be 3 + x squared multiplied by the height. What is our height? That is 9 - x. Are we all good in the hood, guys?

[00:47:23]All right. Z.

[00:47:28]>> Yes, sir. I'm just writing it down.

[00:47:30]Sorry.

[00:47:31]>> Okay. All right. Not a problem. Uh, Valerie Sass.

[00:47:36]>> Yes. So, I'm good. I'm just writing I'm also writing down the question.

[00:47:39]>> Okay. Not a problem. Right now, let's find the volume in terms of X.

[00:47:47]What we just need to do is make sure that we Right. So that's 3 + x all squared. So that will give us 9 + 6 x + x^ 2 right into 9 - x as choice. We need to work this entire thing out. So in this case that's going to be 1 / 3 into right 9 * 9. Right? We're going to use our distributive law. That's going to be 81.

[00:48:25]9 * -x, that's - 9x.

[00:48:30]Okay. And we've got 6 x * 9.

[00:48:39]That's going to be 54 x.

[00:48:44]and 6x * -x will be - 6 x^2.

[00:48:53]And then finally we've got x^2 * 9 that will be 9 x^2 and is going to make sure we are x cubed right okay right right I want to try and so that's 1 over Okay, we've got 81.

[00:49:26]Uh, we don't have any other constant.

[00:49:30]We've got - 9x + 54. Those are our two x terms.

[00:49:40]That's going to be + 45 x.

[00:49:46]That's - sorry uh - 6 x^2 + 9 x that's going to be 3x^2 and finally that's - no not - 3x^2 sorry that's + 3x^2 sorry about that and that is - x cubed right so we've got the product of that entire expression Right. So we need to uh just multiply it by 1 over3. Uh that will be important. So 1 over 3 uh * 81.

[00:50:28]So 81 / 3 gives us uh just need to make sure um am I that's 27. So this is 27 + 15 x + x^ 2 - 1 3 x cubed.

[00:51:05]Right?

[00:51:07]So that is our volume in terms of x. So in essence, what are we going to need?

[00:51:13]We're going to have to find the length of the side. We're going to find the height and then take the ratio thereafter. But such that the volume is maximum.

[00:51:25]Okay. How do we get the maximum?

[00:51:31]>> So don't we get the derivative of the sum?

[00:51:36]>> Very good. The derivative. And what do we do to the derivative?

[00:51:43]So how do we solve it?

[00:51:45]>> Yeah. No, what do we do to the derivative?

[00:51:48]>> We make it equal to zero.

[00:51:50]>> Yes, absolutely. So this is going to be 15. So this is a constant. It becomes zero. So that's 15 + 2x + 3 * 1 / 3 will be 1. So this is - x^2 and this is going to be equal to zero.

[00:52:11]Okay. So there's a quadratic equation.

[00:52:14]Now please and I want you guys to listen very carefully on this. So which means we are going to find two solutions on this.

[00:52:24]But you will see that one of them we will have to get rid of. Okay. One of the solutions we'll have to get rid of.

[00:52:31]And uh in this case how do we solve this quadratic equation? Right? If I multiply everything by negative, that's x^2 - 2x - 15 is = 0. Let's factoriize.

[00:52:47]So that's x and x factors of 15 such that when I subtract them will give me two.

[00:52:55]That's five and three. That's negative and that's going to be positive.

[00:53:01]So x is equal to 5.

[00:53:06]or x is equal to -3. Now automatically can x become -3 and if not why not right can x become -3 can you have the dimension of a square become negative I don't think so >> excuse I'm saying there's a lot of noise here, but no, I don't think so.

[00:53:44]>> Yes, you're quite right. So, in this case, this answer is not applicable, right? Because it's negative. But here's another thing that I want you guys to remember.

[00:53:58]So, if we were going to draw this cubic function, what type of cubic function would it be? Remember our a value, the coefficient of our x cub 10 is negative.

[00:54:10]So it was going to be a cubic function that looks like this, right? Where do we have a maximum value?

[00:54:18]You've got your maximum value at your second turning point. All right. Now, why is that important?

[00:54:28]Because if you think about um this cubic function we are trying to maximize this cubic function represents the volume.

[00:54:39]We get two values in this case even if both values were positive. We now need to decide which value are we going to go with. the one value, the first value of your turning point, which it which would be the smaller value would give you the minimum and meaning if I want to maximize, I would therefore take the second value. I don't know if I'm making sense, guys.

[00:55:06]Does it make sense?

[00:55:10]>> Um, sir, >> y >> if they were to ask us to know, it doesn't make sense. I wanted to say where they asked us to find the minimum volume would we take the negative?

[00:55:24]>> No, no, no, no. That's a very important question. Right. In this case, it would not obviously work because the minimum is where x is negative.

[00:55:36]But there are instances tando where you find what you find you get two answers.

[00:55:41]Let's say x is equals to 5 or x= 3.

[00:55:46]But you can only have one answer that that that maximizes that maximizes your solution. Are you with me now? So how do I test which answer is going to maximize my solution?

[00:56:03]And so this is where this knowledge that I'm I'm I'm telling you comes in with.

[00:56:09]Okay. So I've got three and five. So if it's a graph that looks like this, it means the first turning point will be at x= 3. The second turning point will be at x= 5. Are you with me? So if I was trying to minimize then I would end up taking sorry the x is equal to three.

[00:56:33]Does that make sense? N >> yes sir. Thank you.

[00:56:36]>> Awesome stuff. That's a brilliant question. All right. So in this case remember right we're still on the ratio. So we found the value of x. Now we can find the value of uh the height. We said our height where is it? Our height is 9 - x.

[00:57:03]Our height is 9 - x and that's 9 - 5 which is going to give us 4 units. All right. Now please remember we said we are looking for the ratio of the length to the height.

[00:57:25]So ratio So this is going to be five divided by four. And that is how the cookie crumbles.

[00:57:45]Are we all good in the hood, guys?

[00:57:50]We stretched our thinking a little bit.

[00:57:54]Okay. All right. Beautiful stuff.

[00:57:57]Beautiful. Uh Z, are you done writing?

[00:58:05]>> Yes, sir. Yes, sir. Yes, sir.

[00:58:07]>> Okay.

[00:58:08]All right. Um All right. Let's let's enough.

[00:58:15]Let me just check uh on our YouTube uh if we've got any questions.

[00:58:21]Uh, someone says, "I'm writing paper one on Friday and I've never started to practice this topic." Well, you can join us for our workshop and I was going to tell us uh tell you that. Right. The last thing J that I want you to note is that we've got a workshop that is running for to uh no for Wednesday and Thursday preparing you for maths paper one. We've got another one that is coming up on uh Saturday and Sunday that is for meds paper 2 for those of you that are upgrading that are running. Uh for those of you in metric, you're more than welcome to join us for those workshops. Uh you can either come before or you can watch the videos thereafter.

[00:59:04]Uh it is totally up to you. But um guys, I I really want to implore you to continue to practice. Okay. the one h you haven't practiced this session please get yourself uh acquainted familiar with this section uh so that you can be sure to improve because I'm sure that is what uh you want to achieve that you want to improve at the end of the day so uh ladies and gentlemen oh kumbu NBTs Remember NBTs for those of you that uh need to write NBTs, we've got a comprehensive course that will prepare you for NBTs.

[00:59:55]And we still have our special running I think up until end of the month. And you can also get to a uh or rather use the code the promo code malum 11 and you'll get yourself 15% of a discount for that course. Um it is really quite a comprehensive course. I've gone through everything that you need to know uh to prepare for both MAT as well as AQL portions of NBTs. So the link will be on the uh comment section for those of you on YouTube and you know you can always get in touch with us. All right. Uh but otherwise if you do want to join uh the people that are in our class right now uh you're more than welcome our number 06454719 or info atomic.co.za.

[01:00:51]I like how my class is so disciplined right now. They are listening to everything and thinking, "Okay, do we have to sit for this?"

[01:01:00]Guys, please enjoy the rest of your evening.