yes


good job


that's the final answer


good job Chloe!!


do you need anything else?


idk


let me see


huh


do you have anymore questions?


yea


this


okayyy


2 1/2 divided by 1/5?


tea


okayy


so like I taught earlier, how do you convert this into an improper fraction?


5 over what?


because 2x2 is 5 and 1 is 5


uhhhh


noo


I have to go


oh okay


bye Chloe


bye you are my faver


Hi Alex. Good morning, Alex.


How are you today


Hi.


OK


Do you have any problem that


you need to help you today?


I need help with degrees and leading coefficents


And you have


that you have problem with the degree and leadinguiwe or polynomial. Am I correct?


Yes


OK. So wait a second, OK?


I dont have a specifi  problem so.


So here is the principle, OK. Uh, so you need to memorize only 4 situations, 1234. So the first one is like this, OK.


Um


so here


so this is the first one. So let's say, Y equal to X^22. Am I correct? Yes.


OK. So this one is 22 is actually even am I yes or no?


Yes, right. And here, so here, uh, so that's it. I put in 2 you say, but, OK, so this is what they call the leading coefficient, right? And this one is positive, it's not positive.


Can you follow me?


yes


Right? It doesn't make any difference for the timing here, uh, the value of this one here, you can say that well, Y equal to 4 x 2 here, whatever here as long as it is positive.


that's good. OK. And then this one here is even, right? So we got even and positive. So whenever you have even and positive here, even the degree is even, positive is the coefficient


positive coefficient.


OK, and then even power, then the graph is gonna be the graph is gonna look like this.


You see


Yes. And again here, you can extend further say that well, why here equal to 4 and then X square here plus 2 X and then plus 4, let's say. So this one is the same thing here.


All you have to do is just go ahead focusing on this leading term here. So you do not have to worry about the 2nd and the 3rd 1. So all you have to do is just focusing on the leading term. So this is what we call the leading.


the leading term


Right, so the leading, leading term here has a power of even, right? And then the call is not positive, that it's gonna be, it's gonna be like this.


right, so let's say


let's say if you have, if you have Y equal to let's say 2 here and let's say act to the 4 songs, songs on here. Then here, then the graphics are more or less look like this.


Yes. So this is the first, this is the left, this is the left.


pale. This is the left pale.


and this one is the right tail.


But, so the right and the left tail are all pointing up, pointing up here to the same location pointing up.


Sounds good. OK.


I mean again here that, uh, so you can expand the example further here. So let's why he equal to let's say, uh, uh, 21, OK, 21 is an act to the 6 here. So so what here is going to be more or less going to be like this.


it might be possible there is a different here, but


generally the textbook is always, always asking only the left, the left.


the left pale and the right pale.


The left and right is again pointing up.


Yes, but now what happened here, if you have, if you have, let's say, why here equal to let's say 20


Well, so this time around here, so let's say, so this one is positive, am I correct? So I said this time around here, I'm going to put the negative, negative X square here. So in this case, so this one is even, but this one here is negative, am I correct? And again, I don't care about what is the number, but I care only if it is positive or negative. In this case, it is negative. Am I correct, Alex?


Can you follow me


yes.


Am I talking too fast, Alex?


no.


But


OK, so in this case, remember this is negative, so negative is gonna be is going to be like this year.


So this is the left, am I correct? And this is the right. OK.


Yes, so if it is negative, then can be down. So it's going to be pointing down, pointing down. Remember here I'm still talking about that the power is even. So likewise, you can say, why here equal to let's say minus 11 here, OK.


That's at 11/2. I don't care because I can only if it is negative or positive, and this one is negative. Yes, so we got a here so let's say, let's say the 4 here. So in this case, the graph is gonna be more or less gonna be like this.


Right, OK, so that is if this one is even now. So let's talk about the other way around that what happened here if this one is


odd, yes. So the good example is that once it's gonna be like, Y here equal to. So let's say 2 here this one is still positive, but here this is, this is like a cube. So cube is on, am I correct? And this one is even. So the graph is gonna look like this.


Yes, so instead of pointing up or pointing down together. So this time around here we got a situation in which that they are actually pointing to a different direction. So this one to the top. This is the right hand side, the right


the right pa and it's on his left, yes.


Yes, and this one is the Y exist and this is the X axis.


Yes, and again here, it is also happened this happened like this year. So when I why he equal to let's say, well, so we got 3, OK, that's it A here, that's a cube + 2 X + 1.


OK, so I don't care about the second one and the 3rd 1. OK. I can only about this one because this is what they call the leading term, yes, so you need to focus your attention only to the leading term. That's it.


That what the graph looks like, the graphs coming, it's gonna be, it's gonna be like this.


Right, OK, now, so let's talk about something else. So let's talk about what Taiwani if you have Y equal to let say minus 12 here and then act here so let's say cue? Yes, so this is negative and this one is


oh, am I right? And this is what they call the leading term, OK.


So in this case, what you're looking at is going to be instead of this is the first quarter, am I correct? And this is the 3rd quarter. Instead of first and quote quote on here, then you will be looking like this.


So this is the 2nd codon, and this is the 4th codon.


Sounds good, right


So


so this is the first cordon and this is the


3rd quarter, am I right? But here, whenever it is negative and then out here, so you're looking at, well, the, the 2nd quarter, I'm sorry, the 2nd quarter and the 4th quarter.


Yes. So in summary here, you can just go ahead, uh, write it down on your


note here, class note, something like this. OK.


OK. Can you write it down on your class note here because this is very important. This is a very, very good summary of the idea we just talked about.


OK, so here, so this one here is


so let's say,


this one is all about power.


I'm talking about leading power, leading power, am I right?


Yes. And here across here, I'm talking about the leading coalition.


I'm sorry


I call Abian. OK, so L thankful leading, leading. This is leading power. This is leading cos yes, and here, so we're talking about even.


and here we are talking about on.


right


Like this and here leading coalition here we are talking about positive and here we are talking about negative. Remember here, I don't care about the value. I care only either positive or negative. Am I right?


Yes, so here, for example, even positive, let's say this is the very simple example and in fact in your textbook here that well


uh, the textbook asking you to memorize this one because this one here is


considered as


part of the library of graph that you need to memorize. So the graph is gonna look like this.


Right? So this one is Y and this one is. Now, so what happened if you haven't even leading power, but


negative


leading collab. So we got Y here equal to let's say minus X^2 here. So in this case, OK, so it's gonna be like this.


Am I gonna


OK, so keep in mind here, do you still remember something about the vertical reflection.


band.


you still remember


like, uh, neck vertical reflection. This is actually from here to here is


the case of


vertical reflection.


OK, now, so here, here we're talking about on here, on the second no so we get why here equal to x here cube. Am I correct?


Yes. So this one is odd, am I right?


here and then as far as the coy is concerned, is a positive. Yes, so we are looking at the glove that looks like this.


Right, so this one is white and this one is ac.


right? And here what happened here if you have that


the power is on, but the cofi is negative. It's minus X cube here, then the graph is gonna look like this.


Sounds good, right, so you have to memorize this one then, because this summarize everything the concept about polynomial with respect to the leading power and leading coop and positive, negative, or even. Sounds good.


Do you have anything else, Alex?


nope.


OK, so I hope this session is very useful for you.


One question, Alex, if you will, I'm just curious here. Are you logging in from your school or from your home?


home


OK, so you are not going to school? Are you homeschooling?


no


OK.


OK then. So I assume that you are homeschooling with your parents?


No i go to school.


OK then. OK then. So if you don't have anything else, then I'm going to close this session again. I hope this session is very useful. And then here, I wish you a good day, a successful day today in your school. Thank you so much and say hello to your parents.


Bye


Bye-bye


Hi.  How are you?


I'm fine and you?


Good.  How can I help?


I want to work on related rates stuff cause i'm having a little trouble with it


Ok.  I am going to put my mic and speaker on.  You can if you want to or are able to.


All right, I've got my mic and speaker on


helps me free up a little bit


Do you have a specific example of a problem you would like to work?


Yeah, I'll put on the board.


Can you see it


Yeah, I haven't seen it yet, not yet.


It's not showing up


Not yet. I'm looking


not seeing it anywhere


Sometimes it takes a little while to load.


Mhm


There it is


So let me read this


OK


So I'm getting a picture of a rocket going up.


Camera 5000 ft away.


when the rocket 1000 ft above the launch pad, its velocity 600 ft per second. Find the necessary rate of change of the camera's angle.


as a function of time.


so that it stays focused on the rocket.


So


do you want to try to draw a little picture of it.


or would you like me to take a shot at it?


Yeah, I tried to picture part.


You said you already did that


No, I said no, you can try it.


OK. So here I go.


All right.


I'm having trouble. Hold on, please.


Nope.


All right. It's very crude.


All right, so


here's our angle theta.


Uh, we'5000 ft away.


and


at the moment


that we're looking at it


the rocket 1000 ft above the launch pad.


It's up there


and it's going up 600 ft per second.


So we could say that the tangent of the angle is always


the opposite


over the adjacent


Yeah.


All right, can I stop you for a second?


Yeah


Yeah, so basically, in the drawing, where is the camera?


The cameras


right over on the right here


OK. And where's the launch pad?


Launch pads right over here.


on the left


OK. So


when it says a camera is positioned 5000 ft from the launch pad. That means like from the right side or left side.


It doesn't really matter. I drew it, I drew it on the right side, but it could be on the left side.


It really wouldn't matter


Either way would be kind of a right triangle, right?


OK.


So you're, your cameras were, we're not gonna, we're gonna assume that the camera is on the ground and it doesn't have any height.


right


Mm


So we've got a little right triangle. I put it


on the right, the camera, but it could have been on the left.


and


at any moment in time,


if we think about the tangent of that angle


that would be


the way to


the way to try to track


um


the, the rate of change in the camera's angle.


We would want to come up with something like the theta DT.


OK


Does that make sense


Yeah, and then, um,


Well


looking at it without the like data stuff, I might just assume that uh the formula for this one could be, uh, the pothagram theorem so a^2 + b^2 + c^2. Like, how did you come up with the whole data thing.


so what we're, we're asked for, we are asked for the rate of change of the angle, right?


So


I'm not really sure that the Pythagoras theorem


is, well, we'll have to see, but


if we could come up with a, a formula for the tangent of the angle.


as a function of time, then we would have it.


So, it may be that the Pythagorean theorem comes into play here.


but not necessarily.


um because we really can't


do much with the angle with the Pythagorean theorem. Do you see that?


Yeah, I'm just trying to understand like when do I know to use the thagran theory and when do I need to use like tricks.


Right. So let's just say the problem was how fast is the distance between the rocket and the camera changing, OK?


OK


That would be the more typical problem that you would see is, you know, the rockets going up


and it


and 5000 ft away and the rocket's traveling 600 ft per second and it's going up and up and up.


Mhm


then definite, so if we were asked how fast is the rocket's distance changing from the camera, then we would definitely use the Pythagorean theorem.


to come up with that, that distance.


as a function of time. But in this case


they're actually focusing in on the angle, and I'm not so sure that the Pythagorean theorem is really gonna help us.


Uh, we may need it, but


if we really wanna solve the problem. I think we have to focus on the angle this time, and it may not be


um,


able to be expressed


with the Pythagorean theorem because that really just relates the sides of a right triangle to the hypotenuse.


Oh, so like by camera angle, they mean like right here.


I'll mark it


Wait, let me show you, like right here, right.


Yes, that's it, Theta. Yeah, I should, you're right, exactly.


So


yeah, because


OK, now it makes sense. Since they're like showing you that small part, that's why if they're talking about this like whole line right here, then it would have been like the pathography.


Yeah


yes, if they were saying how fast does the distance between the camera and the rocket changing, you would definitely, and that's the more typical problem that you would see.


Um,


OK.


you nor more typical problem would be something like that.


So,


but it, but unfortunately or unfort unfortunately, it's not that simple when you talk about the angle, but you'll notice that


um with these problems, when they talk about angles, usually trying to use the tangent is pretty helpful.


because


um if we can express the tangent of that angle.


as a function of time that we pretty much have it


Mhm.


So, I'm gonna think about this for a minute, but the tangent of the angle is always the opposite.


is always the opposite over the adjacent side, right?


And we know the adjacent side is always going to 5000.


OK


feet because that's not gonna change.


Yeah


So now the question is, can we come up with an expression for the numerator or the opposite side.


And we know that it started out 1000 ft.


So I'm thinking 1000


um


right


We


to kind of


Are you writing down the board?


I will in a minute. I'm just kind of talking, talking through it right now. So I'm gonna write it on the board in a minute.


OK.


Um, so let me go ahead and do that and we'll see what we think this is right. So I'm gonna go down a little further now.


and please stop me because, you know, there's always


many chances for error.


But let's go ahead here


Mhm


tangent offain, I apologize for my handwriting. I've got a


OK


kind of a tremor thing going on. So that's always the opposite.


over the adjacent


and the adjacent is gonna 5000. We'll keep it in feet.


And the opposite is always gonna 1000 + 600T.


Oops, no, I have to reset.


Excuse me here, let me see if I can do it again.


No


I'm sorry


One more time with feeling


So


Let me lengthen this


I can't write very well, 5000.


So if we think about it


iste as the number of seconds that go by. We started at 0.


then the tangent of the angle is going to be 15%, right?


1000 /.


Yeah


As, as each second picks by.


the the tangent of the angle is gonna get bigger and bigger.


Mm


um after 1 2nd, it'll be 1600 over


5000. 2 seconds, it'll be


/5000 and on up, right?


Does that make sense to you


You mean?


I'm sorry


You mean? What, what did you mean?


OK


So let's, let's just look at this expression.


tangent of the angle  = . I, I'm sorry about my writing. It's 1000 + 600T / 5000.


So at time equals 0


Mm.


if T is equal to 0, 600 times 0 is 0, and we would end up 1000 /, 5000, excuse me.


Mhm


If it at time T equals 1 2nd, we would be 1600 /


5000, right?


Yeah


At time T equals 2 seconds, we would be at


2200 / 5000, and so forth.


Yup


So we have a, we now have an expression, if we've done this correctly


of the tangent of the angle


in terms of tea


And that's pretty good because now we can use our


knowledge of derivatives


to find the theta DT.


All right, real quick, can we, can I ask, wait, why, why is it plus and not like times?


It is time


but we're starting out in


Don't 1000 + 600.


Yeah, I'm sorry about my writing. I really apologize.


It's a 1000 + 600T.


Can you see that or


No, I can see that I'm just saying like why is it that way?


OK


So think about that angle


Mhm.


If we're just starting out


do you agree that the angle, the tangent of the angle is 1000 / 5000.


Yes, I agree


OK.


If you, if you plug in T equals 0 into the formula, we would get 600 times 0, right?


Yeah


And we would end up 1000 / 5000 when we're starting.


After 1 2nd


if we just choose an arbitrary value 1 2nd.


what would the numerator be?


if I plug in 1 2nd


led to 1600


adding to 1000, yes, 1000 + 600 times 1 2nd, right?


So we would end up 1600, correct?


Yeah, I'll get that part. I'm just saying like the 1000 + 600. Like why is 1000 + 600 and now 1000 times 600.


So that


we're, we're gonna


or minus 600


we're, we're, the rocket is going up.


Mhm


We have to multiply by the 6 6001 times the tea, if that helps clear that up. So,


Yeah, that part is clear


OK.


So


if you think about where the rocket is vertically.


We can't multiply 600T. It's starting out 1000 ft.


Mm


So we always just start out 1000 ft.


and after what, after 1 2nd for for example, we will be 1600 ft. Do you agree with that?


Right


and the way I got that was after 1 2nd


the velocity 600 ft per second.


times the time


is 1 2nd


So that's 600 ft that we would have after it would have climbed 600 ft


after 1 2nd


Mhm.


Yeah, I see.


climbed 600 ft above 1000.


OK.


Yeah, perfect.


So as we can, we can see that the expression works.


for any time T after t equals 0.


if we had at 2 seconds


the rocket should be 1000 +


600 times 2 seconds.


which would 2200 ft.


so we would 2200 ft where the rocket is at 2 seconds.


over.


502,200 divided by would be the tangent of the angle.


Yes


So I, I detected


I


please ask again if I'm, if it's not clear, um,


No, it's really clear


so I think we've got the hardest part done here because


now


we have an expression for the tangent of the angle, and it's in terms of time


So what we can do is we can


use


to differentiate both sides of this equation by


time.


and we should be able to get how that angle


changes with respect to time.


OK


Hey


we ask more questions if, if I'm


if it's not clear


Yeah


So now


um what would be the


so if I'm differentiating the tangent of theta, what would I get if I'm differentiating that with respect to time.


If you're what


So if I take the derivative of the tangent function, what would I get?


What, what kind of


Um,


uh, 2 square.


beautiful


So, if I can write without


messing this up here


you're exactly right


We would have


sink and square theta.


and by the chain rule, since we're


multiplying, we're differentiating with respect to time. We must say the theta DT.


Mhm


So we've, we've taken that


derivative with respect to time on that side. And now we need to do the same thing on the right side.


And if we do that


we actually see


we could use the um


see the best way to do it


Several ways to do that


We could just, um, simplify this as


if we wanted to, probably the easiest would be to have it as 15th


We want to take the derivative with respect to time. I'm gonna have to raise.


All right, we've got an expression which has a denominator 5000.


So,


I want to take the derivative with respect to time.


If I do, I can break this into two


variables


The first one will be just 1/5.


and the second one would be


um


I


6600 over 5000, which would be 6/50.


times T


So all I did was just divide


I took the numerator and broke it into two parts.


say you have to brush your teeth.


So far, so good


Yeah


OK


was my teeth


All right.


Now if we take


Sorry about the


oh, you're fine


All right, let's just rewrite what we have on the left. Sekin square.


beta


the theta DT.


and the derivative of a constant is zero. What would I get for the derivative of the second term?


Um, you would get, wait, let me see. God bless your teeth.


What do you want?


That's it


OK, so you would get um 6/50 and then


yeah, the DT well the time?


Say it again, please


Yeah.


Wait, is the tea the time right?


OK.


So


what we have is we're saying sequin squared theta d theta dt.


equals


Mhm


right?


All right


Um


now


the question is, how could, if we solve for the theta dT, which is what they're asking for in the problem.


Yeah


we


um, we, we would take 6


um


so hold on a minute, let me think about this.


So we still have to divide by


we saw for uh defeated DP.


um, seeking to


one over cosign, right


Mhm.


So we would have


so now the question becomes, how do I find out what sequin squared theta is.


What


So we can't let our um answer with chicken squirt. Oh no, because you want the DDT on its own.


right, exactly


Right.


So now the the problem we're facing is what is sequin squared beta.


Um


Yeah


it, yeah.


well, it


skin square theta times def theta dt is 6/50.


I


Um.


so I've, now we need to think about what sequin squared theta, if there's anything we can do to get that to some kind of


expression


so, let me think here a minute.


Yeah.


So this may be where we might have to


use a little


Pythagorean theorem. So,


take is


one over cosine, right?


So


the cosine squared of


So if we saw for the fatedta DT.


equals 6/50.


times


cosine square


theta


Wait,


Well, so what did you do on the left side? Did you divide by 2 square?


Right, exactly right.


So you divide it by, OK, you divide a second square and you divide it on the other side.


So if I divide both sides by seek and squared.


Yeah


I get


the theta ET equals 6/50 times the cosine squared of theta.


right?


If I haven't made a mistake


um


Well, let me think about it for a second.


Sure, let me think as well because we need, we need to make sure.


Yeah, cause 6 or 50 divided by 1 over cosine is um


5000 times co-signed over 1.


So it's basically 6500 co-sign.


co-sign


Cosine squad


right


So now this


So, yeah.


is, this is starting, starting to come back to what you were talking about. Do I need to


I, do I, what, how am I gonna figure out how, what the cosine square


is doing


Um


so let me think as well for a minute here. Divative tangent sequin square.


faded deeply


But before you think for a second, I just wanna know why are we like changing it to one of our cos I, we can't just keep that again square.


Well, if you want to, you could use 5000 divided by sequin squares, right?


Oh, but it's just like the vision, right? That's why we don't want that.


Right, you, if we divide both sides by sequin squared theta.


then we get defeta DT on the left.


divided by sequin square


which should be cosine square.


OK


Correct me if I'm wrong


1 over cosigners seekin, right?


Sometimes I get that


Yeah, it's.


All right


it is.


Um


so we, we have a problem still because we don't know what cosine square to theta is as a function of time.


And


it would be nice


and we should be able to


we need to really


get that as a function of time so that we can finish the problem, right?


Yeah.


So if we look at that angle, the cosine has always got an adjacent side 5000.


so that, and then it would be over the hypotenuse, which is


changing according to Pythagorean's theorem, right?


Mhm


And that would be


the hypotenuse of that would be the square root.


I'm just thinking out loud here now.


of 5000, I 5000 square.


plus the same old number we had 1000 + 600T


times


T


So I'm gonna write this in here for cosine squared.


cosine


beta


and then I'm gonna write it for cosine square data.


So just bear with me for a minute here.


Yeah, no problem


So I'm gonna write cosine theta


over


the square root of 5000


square


plus


squared


Do you see that


OK


If you can read my writing


I can see.


I'm gonna fix this part here, it's too messy.


Mhm


We're almost done


Yeah


All right


as good as I can do. So do you see that the cosine theta


is always gonna be 5000, which is the adjacent side over the hypotenuse.


which will be using the Pythagorean theorem.


5000 square plus, that should be (1000 + 600T)^2, that should be (1000 + 600T)^2, shouldn't it, not a 1000 tea.


I'm sorry


That should be


that I've, I've got, I haven't completely written the expression properly. I got we race here.


I've got a


Well, would that be (1000 + 600T)^2


or would it just be


it would be the same thing that we used before, 6


1000 + 600 tea.


1000 + 6600


Yeah.


So let me just kind of write that


up.


Yeah, I need to erase.


I do apologize. I, I have a problem writing.


and


No product. It's not a problem.


but I think, I think the reasoning is all solid that we've, we've got here.


All right


I don't really have room, so I'm gonna need to.


waste a little bit more


Yeah


It's it kinda looks complicated, but it's really not something that we're gonna have to simplify it all.


cause we'll just plug this right in after we square it.


So this is gonna be


this is gonna be 5000


No, it's a 1000 + 600T, right?


Mm


Um


+ 1000


tea.


square


It's 1000 + 600T, isn't it?


Yup, it is


I can't turn to write that properly. Let me try one more time.


Do you want me to write it for you?


If you can


it would be helpful


OK, I'll write.


I


Since I've done it wrong about 3 times now.


It's OK


Yeah, that's much better


Great.


So we now have an expression for cosine of theta.


Mhm


and it's based on the same logic that we were using for how tall the


you know, the rock, how high up the rocket is at any time. It's the same thing.


But we now have an expression for cosine theta and thus we have an expression for cosine squared of theta.


which we could just square the numerator and denominator.


we would get


5000 squared on the numerator and we would remove the radical sign from the denominator.


and we would be done


So we would have the complete expression for defatedta DT.


6/50 times


what I just said


cosine theta squared, which is that mess there.


squared.


Also, everything's squared, like the whole 5000 over those things square.


Yes, so what you would do is


I can try to write it


or you can


Uh


right


Since that's cosine theta.


cosine squared


which 5, let me see times pounds 5000 is, make sure I don't mess that up.


5000 squared is


a big number


25 million


might be best just to leave it this way.


Leave it 5000 squared over, and we would simply remove the radical sign from the denominator.


right?


Yeah


So our final answer, um, I guess I'll try to write it here.


See if I can do this without messing up too bad.


The beta DT


is


times


Mhm.


5000 square.


over


plus


P


where


So that's our answer


Mhm


Now express the angle as a function of time.


at 6:50


fraction 5000 squared / 5000 square + 1000 + 600 t squared.


I don't know if you have


an answer somewhere, but


yeah


Yeah, there's the answer key


So whether it looks like that or not, I guess we'll find out. Um


OK, let me go check.


OK, looking at the answer, he, it looks like they did, there's a little bit different than how we did ours.


and which like it's really a lot, but like that ends the insure that they got was 3/26. rats per second.


So we need


I can


yeah, I can send the thing on here so you can look at it.


OK.


um


so they did a radiant per second and they actually came up with a number, right?


Yeah, they came in


So


what


So do you want me to send the thing here so you can see like the steps that you.


Sure, um, what I don't understand on their answer though is


the angle changes as time changes and in the, in the question it said


as a function of time, OK?


Mhm.


But you're, you're just giving me a, a number


one number for radiance per second, right?


Yeah


So that bothers me because


the angles changing as a function of time. If they said after 1 2nd or after 2 seconds what, then I could understand, you would, we would plug in and we would get radiants per second.


But


I don't, I don't see any problem where they um


said what time they wanted to have


um, so


anyhow, yeah, I'd be happy to look at it, but I'm


I feel like there's a


little piece of information missing like at what time do they want to see how fast it's changing.


anyhow.


Um


OK


wait, let me just send it to you so you can look at it, OK.


Is it gonna be below all the stuff we wrote or?


Yeah, uh, it's probably gonna pop up up on top.


OK, whatever


Yeah


Hopefully


Actually ended up being below


OK, that's fine. Let me get down there.


Oops.


OK, so


um


so, so they did, they did set it up a little different. It's the same idea.


Well, can you see or do I need to like zoom out a little bit.


I, I can see it


I can read it OK


Um


so they're saying


the tangent is the high.


and


they did the same thing we did.


so I


see what they did


OK, notice that when they


they said when when H equals 1000.


when the height 1000, which is at times 0, OK?


Mhm


So they're actually, they're actually taking


um


they're actually kind of setting the time equal to 0, if I'm, when I'm looking at this,


So we missed that part


I think they missed it, but let, let me just say why I think that


So the problem


as you put it up there at the beginning,


it said


let me go all the way, I have to go all the way back to the original statement, but


um,


OK


this is


in my opinion,


this is actually just showing it at that initial 2 at time equals 0.


Um.


but it doesn't answer the question as, as


when time equals other values, OK?


Mm


Do you see that


See what I'm saying


Yeah


The question didn't say what was the rate of change at time equals 0. It just said come up with an expression


for it. So if we actually plug time equals 0 in


to our


um


we should get the same answer


Do you see what I'm, I'm questioning there.


Yeah, I see what you're trying to say.


Because


I'm


it said come up with it, with it as an expression of um


with respect to time


Well


it's fine if we plugged in t equals 0.


then it just becomes


5000 squared / 5000 square, doesn't it?


Um


no, becomes, becomes 5000 squared over


5000 square + 1000 square.


So all I'm saying


is


their, their answer is right.


for


if you're just looking at the point where the rocket's just starting the launch, that the rate of change


going to be at that point


But if you wanted to find out what the rate of change is of the angle at other times, it would be a different answer.


Go ahead and plug in T equals 0 into what we did at the very bottom and see if I get the same answer as they did, OK?


OK


Do you, do you see what I'm saying?


Yeah, I see what you're saying


OK, so I'm going to


get my calculator out here.


I'm gonna do a fraction


the 2nd 1, 2nd fraction first there.


I'm gonna use 1000 square.


divided by


5000 square


plus


and since time equals 0, it would be 1000 square.


So 1000


square


and I get 25, 26, which is exactly


well, let's


so 25, 26 for that answer times 650.


So I'm gonna take times


and


so


our answer is exactly the same as theirs


when you plug in T equals 0.


I, I worked through the math, OK?


You get, you get the exact same radiance per second.


Are you still there


Yeah, I'm here


OK. So if you plug in T equals 0 into what we did.


It gives it, it gives the same answer they get.


OK


But this, this answer that you and I worked on is


far better


because it allows you to


find out the rate of change of the angle


at any time after


the launch


Sure.


Their answer is only for the


for


at the instant of the launch, how fast does the angle changing?


But if you wanted to find how fast the angle was changing after 1 2nd, you would have to plug in one into our


formula.


If you wanted to find out how fast the angle is changing after 2 seconds, you would have to plug


2 into our formula and so forth.


OK. Here's how I see that the, maybe the problem will also saying that already, but we just maybe didn't catch it because it says when the rocket 1000 ft above the launch pad.


find the necessary rate of change. So maybe that when.


Ah, you're right


is when you're trying to tell us, yeah.


Very good, very good. So


I, I still like our


I still like our solution though.


Yeah


because it's more general, but you're right that I think you've gotten the reason why it's at T equals 0.


So,


yes, it would have been easier to do it their way.


but I find


our way or this way to be much more gratifying or


um helping and understanding, because if you think about it intuitively, you know that angle is going to


increase


um


you, you know the angle's gonna be changing as time changes, and we came up with an expression


that


gives the rate of change of the angle with respect to time. That's radiance per second


But.


Mm


this, this allows you to, to find it at any time, not just at the launch time, but


anyhow,


um, I hope


I hope I haven't led you into too much extra time or work on this, but


um,


I, I think both answers are right


Um.


they get to it quicker


because they didn't really


do a general solution to it, but


anyhow.


Yeah, no problem.


Do you, do you


follow everything we did because if you did, you're doing great


Yeah


Do you, do you have time or you gotta go.


I have time. Um


do


so, um, getting back to your original question,


you know, how come it


we didn't use Pythagorean serum.


We did a little bit on the


on the cosine part, but


again


for this problem, the main trick is to, to look at the tangent of the angle.


and do


be able to solve the problem.


Sorry, give me one minute. I'll paste the next one.


OK


And uh


Do we want to clear the board altogether?


excuse me


I can erase it


Do you want me to clear the board altogether or do, do you have what we did save some


Yeah


So would you like me to do just clear the board altogether a whiteboard?


Yeah, you can


OK.


There you go


Can you see the next one


Yeah, I see it. Thank you


All right. So,


Wait, do you want me to kinda like start it so that like to make sure that I'm getting it because I have a test on condition


and you can


yeah.


me and correct me and stuff


Yeah, uh, it is better if you do it, especially since my


penmanship and etc. is awful. Why don't you draw a picture of what, why don't you draw a picture of a cone upside down.


that will be the good first starting.


Mhm


just a nice cone upside down, would be great.


I would be nice, but


yeah


Do it as a side view, just, you know, like a, uh, you don't even need, well, you can do it that way, whatever.


OK, that's good


Um.


Wait, what do you want me to do


you're, you're OK


Um


OK


all right


So, water is draining around the height of the funnel is 2 ft.


OK, so the high, so this right here, right?


Yes, that's exactly right.


And that would be 2 ft.


And


and the radius of the top is 1 ft, so we can, we have a 1 ft radius.


OK.


and


So at what rate is the height of the water funnel changing when the height of the water.


So we want


So we're trying to find the DH


B T right


yes


When


it is one hat.


OK


So I think the key to this problem is probably gonna be something that might surprise you like similar triangles.


Um


so


if you think about it


um


Oh, we're missing something. So what are you gonna not do it. So we have that 0.


03 ft.


This looks like it could be the volume.


the water is draining from the bottom of the cone at a rate of


cubic feet per second, right?


Is this the volume


That's volume, yes


So the volume


Yeah, cause I thought you.


so what, what, let's see, the volume of a cone, you can help me out here, is it uh 1/3


1/3 pie


R2H, I believe, 13R2 H.


Yeah


Right


Something I need to like remember and learn, or it's definitely the formulas of these like shapes cuz I really have no idea like on any of them. I don't think they're gonna give us a formula sheet, so you have to remember all that.


Um


I'm


I'm pretty sure the volume of a cone, I'm gonna check


give me a minute to check for the formula. I'll be right back.


Mhm


OK


Well, so it's, it's, what did you say it was?


It's out, I'll write it out there for us.


try not to mess it up


up too fine too fine here. So,


volume


is 1/3


I


H.


So that's a critical


critical formula for us


What does it look like H has some feet and it's working.


because I'm really terrible.


No, it's OK.


But thanks for the humor. I, I need it.


Yeah.


OK, so


we can see that um


BV


so we know that DV


DT


cause


Mhm


negative


feet per cubic second.


DV


So that's interesting information. And so


um,


if we're trying to find the rate, the height of the water is changing when the height is 15. We could solve for


H


Well, let's think about this


If we took the derivative of both sides of the equation with respect to time.


Do you want me to take the derivative of it?


Well, not quite, but I'm still kind of thinking for a second here.


So,


um


so we have one problem


Not only is H changing


but R is changing as well.


So if you think about it


when, when the water starts going down,


Mhm


the radius is, the radius is 1 ft at the top when it's full, right?


But when it, when it goes down, not only is the height changing


but the radius is also changing.


Right


So what we need is we, we can't just go willy-nilly off into the


we, we need to have, be able to account for that.


in our


in our calculations.


OK


Well, one little trick, one little trick here is


that if we look at the side view of um


give me a minute here to kind of work this out.


But if we looked at the side view


we know at the top


it's got a radius of


1 ft


When the height is


2 ft


but when


we're at any other time


uh


or height


it's going to be different, and there's a relationship between


the height and the radius.


And once we get that, we'll have, we'll be able to solve this problem.


because again, the radius is changing


and


what we can say is


is equal to


R over H


I'm gonna write this down right here.


You'll kind of see what I'm talking about.


So if I, I'm not gonna destroy your nice drawing, but


if I have a side view,


of this cone


we know that the radios up the top is one.


when the height is


But when we're at another


height,


the radius is something else.


say 1.5


Mhm


So it's a little bit of a proportion.


Um.


the radius to the height is always R over H and because


we have


kind of similar triangles working out. We can always say that the radius to the height


will always be 1/2.


You said the radius would always be 1/2.


The radius to the height, the ratio of the radius to the height will always be 1/2.


So if you look at the, if you look at the big triangle


or the, the big triangle, the whole cone looked at from the side.


Mm


We know that the radius is one


over the height is 2.


If you look at


when the water's lower, the radius is going to be something else.


Mhm.


and the height is gonna be something else, but they're always


they're always gonna be in proportion of 1 to 2, because they're corresponding parts of similar triangles.


Do you see what I'm saying


Yeah


So, if you look at, look at the whole big triangle, which is the cone looked at from the side.


The radius is one and the height is 2.


When we look at


when the water's lower


the radius is gonna be something smaller, and the height's gonna be something smaller, but they'll be in proportion of 1 to 2 because they're


it's a similar triangle to the one that's


the big one


OK


Do you kind of see what I'm saying


Yup


So if you believe in your geometry, which you


I'm sure you had at some point. We know that corresponding parts of similar triangles are in proportion.


So


because we look looking at the side, we can see that


the small triangle when the water is lower


is a similar triangle to the big


cones


triangle, the, the when it's filled up with water, and the ROH will always be 1/2 in this case.


Mm


So that is the trick to making beer this whole thing work because if we didn't know how the radius changed.


with respect to the height, we'd be missing a critical bit of information.


So what we really want to do


just bear with me a little bit more here before we, we're ready to solve this.


We can solve for our


I think ours is like 14.


so 2R


hold on a sec. 2 R equals H.


So an R is


H over 2, right? Oh, we don't want to sour our, we want to sulfur H, so.


Yeah, 1 is over 8


So H is 2R.


The height is always 2 times the radius.


but R is alwaysh over 2 as well.


So if we substitute H over 2 into our formula for volume.


You're right, you


will get one, like a 1/4th factor. So what we wanna do is we want to take H over 2 and plug it in for RR.


So we get volume equals 1/3


I.


H/2^2 will be H^2 over 4, right?


I think before like plugging stuff in, we need to like differentiate it and then we can plug it in at the end.


Um


well, you only want to differentiate with respect to 1 to 1 variable. You don't want to have two of them like R and H, OK?


You


OK


you see what I'm saying


I'm gonna let you write this out cause I'm not doing very well


So what I would


Mm.


what I would like if you would do for us and then you can


continue and finish the problem, which won't be too bad.


is what we want to do is we want to say,


V equals 1/3 pi R2H, but instead of R, we want to put H over 2 in there.


Well, you said to do what, like B equals 1/3 pi.


So


instead of R, we put


H instead of


So if you could write that out, that would be great.


Mhm.


just below my stuff or whatever.


I


We was the age over 2.


OK, let's get back to the side view of the cone.


So,


I thought it would be 1/4.


with R


well, it will, you put an, OK.


if you put


we saw, we saw that there are similar triangles


Mhm


The radius to the height will always be 1/2.


right?


Yeah


Therefore


the radius is always half of the height.


So just plug in H/2 for R


So instead of 1/3 pie, you got 1/3 pie


Now, x will be


R^2 over 4.


So the next thing after your pie, put in parentheses 82/4.


Yeah, but we


So we don't wanna plug for HA? Is that what you're saying? Cause we already know what H is. We're trying to find age at 15.


yes, I understand, but we need to get the right, we need to get volume as a function of height so that we can differentiate


If we don't plug in for R right now.


So r over 2 square.


right? Just put a little square good, and now put


H again


after that


Now, stop right there because that's beautiful.


We're saying that the volume


in this particular instance is as a function of height, and we can simplify this a little bit, but it's 1/3 pie, R2H.


But we now have everything in terms of age rather than R If we try to differentiate before we do this,


we won't, we'll have a much more complicated problem.


because we'll have several variables. We're only interested in volume as a function of height, not as a function of the radius.


Am I making sense there


Yeah.


So let's simplify that. Why don't you go ahead and write it as B equals 1/3.


high


below there, just write V equals 1/3 pi.


Mhm


and we'll have H cubed


over 4


and we can really say that and just put another set of equal signs.


equals


high over 12


H cubed


So


Pytol


by over 12


HQ


OK


Now


I, I know that was a lot to go through, but if you followed it,


you now have V as a function of H only, as opposed to RNH, OK?


Mm


If we try, if we try to do it with the R in there


we would be


unable to solve the problem


All right.


So getting back to the


now we can go ahead and start differentiating


We can take DVDT, right?


I'm glad I've got somebody who can


use the board.


So


Mhm


what will we get when we differentiate H with respect to T.


OK


Um


3 pie H^2.


Good.


DHDT.


DHDT


Does that make sense to you


Yup


Now


we know that we were given DVDT.


I forgot what it is, what was it?


Point


Uh 0.03 ft.


Yeah, let's, let's right below that -0.03


for DVVT


equals 1


we can say 1/4 pie.


312 would be 14 5.


before, whichever way one.


H^2 DHDT


But can we just not substitute already.


Well, I


I


um I don't know, I don't, I don't know about that.


but


Oh, OK


you can't, you, yeah, you can, so that, that's one, it was one half, right?


Yeah


They


So you go ahead and put 1/2 squared, right?


Yup


OK, beautiful


So square


and then DHDT, right?


Beautiful.


All right, let's


let's solve, uh, simplify a little bit more, so negative point.0.


03 equals


looks like pie over


(1/4)H^2, right


Time DHDT


Mhm


OK


we should be able


we're asked to find, we're asked to find DHDT. So,


let me go


So we divide the pyro


Yeah


Actually we're multiply 16 over pi.


right


to both sides


exactly right. I'll go ahead and do it on my calculator. You might want to go ahead and do it whatever way you wanna.


and 0


OK, I got an answer. Let's see if we get the same.


I don't think we need to plug for pie.


OK, um, you can do that or not. I did plug it for pie and I got


negative 0.15


but depending on what your answer key shows, um.


Yeah, the answer key uh shows -0.48 over pi and it shows equals -0.153 ft per second.


that's exactly the same thing I got.


Awesome


Awesome, awesome, awesome


Very nice job


Right


So


you might say


you know, the, the I, the only reason I know


that I had to get rid of the R


is because of experience.


So there's always a little trick to these problems.


In this case, it's similar triangles.


where you're trying to


get rid of the R variable so that you just have things in terms of age.


Mhm


but it's not really


necessarily gonna come always without experiencing the, you know, these problems.


just working them. That's like the tangent of the angle when we did on the last problem.


You know, you're only gonna know to do that after you work a few and you say, oh yeah.


but


anyhow


OK, I had a question about like the units cause we definitely have to put you in my gift points up. And the last one, it was Radiance per second, but like on the


Yes.


on the problem, there was nothing like that said that ever said radiance. So does that mean that for every trach functions like we use like the answer always have like radius for like seconds, minutes, whatever.


OK


So when you're dealing with angles


and you're dealing in calculus.


Radiance is a sort of a natural unit of measure.


that


is used, degrees do not mean anything in this, in this world.


So whenever you're, you're solving for something defeta ET or, you know, just solving for an angle, and you get number, it's always gonna be in radiance.


It's sort of the natural


measure of an angle


Um


so


yeah, it'll always be, when you do get answers, you should plug in pi and you could always get answers in radios and


degrees do not


ever come into play really, unless you're just converting from one to the other to give it in, in that


angle.


measurable like degree measurement.


OK, so do you still have time or you gotta go?


Uh, I got time for a little more.


I, I hope


you know, you're kind of getting


getting the idea. It's really not that difficult, but like I said, there's


OK.


tricks to every one of these, and that's the hard part. Don't know what the trick is.


OK, let me go on to something.


I'm gonna


take a break for a minute and I'll be right back, OK?


Yeah, you can take a break


I'll be back in about 2 minutes.


OK, no problem


OK, I'm back


All right. Can you um delete the board?


Sure


OK, so I found the like cause I told you I already have a big test coming up like give out, um, related rates and um global minimums and maximums and stuff. But um since we're, we're working with like related rates today, I found like a midterm, like a, like a sample.


Uh, so


OK


but he has like different parts to it. So this is like the first part. It has like the


Yeah


question and he has like stuff that we have to go through like questions you have like


letters A to


Go


E. OK, but they're not like a lot, so.


I'm, yeah, I think it, it seems similar.


for the first one to read and I read.


I'm sorry, say again


No, you can go ahead


Anyhow, it's, it's not unlike what we were just doing.


from the looks of it


We're, we're


Yes


it's, it's very similar to the problem we just


did, and they're all kind of have a theme to them.


So let me see what this one says.


Mhm


Right


OK, but on this one, since, uh, I kind of want you to like, I kind of want myself to go through like everything and then uh whenever like I'm make a mistake, then you can be like, oh, that's not correct. Uh, and then if I have a question, I'll just ask you, but I really want to go through all of them.


Yeah


Yeah, definitely, uh, definitely.


So, I'm just gonna read the first part here.


Yeah, I'll read it


OK.


I'll read it out loud. The cylinder is changing size in such a way that the volume remains constant 12 pi cubic meters.


Denote the radius of the cylinder by R and the height of the cylinder by H.


and the area of the circle at the top of the cylinder


by


OK


kind of stops there. That's kind of weird.


by A. OK, I see it.


All right.


So


I'll let you go ahead and


go ahead and uh


start working it


OK, so for letter H, turn this back, it says that uh live with the diagram with the length R and H so draw something on the diagram.


Mhm.


It's kind of small here but


yeah


OK it says RNH do I also have to like add the volume, I mean, the the values or no? Just keep it at that.


Say again, ask that again, please.


So I'm saying like it says labels with links R and H, so doesn't want me to also add the


the, the, the values of the age.


Um, I don't think so.


at this point


Yeah, since you said RNA.


Yeah


So you did that just fine, yeah.


OK


OK, I'll just write what we know because it makes it easier. So we know the volume


Oh sure.


By the way, you can stop me whenever I'm doing something wrong.


So


OK.


um


what you're doing is fine, but if you're going down to point B, I


you know, that's what the next step they tell you to do, right? So.


Exactly. That's great


Yup


Yeah, I'm just writing it down so like I have it in my head and I can see I can picture it cause that's what the pro uh it uh the teacher also tells us to do.


Mhm


OK, since I see that we have the like the volume. I'm guessing that for the formula, we're gonna have to find like the.


something um either with the volume or the area cause we're trying to find the


area, but we had the volume, so maybe you need to find like both um formulas and like introduce it, if you get what I'm saying.


Yes


OK, I for I forget what the um formula of the um volume for the cylinders. Is it H times


uh


it's.


pi R


I heard you say it


2 times age.


So think of it as slight, think of that as a as a piece of cheese around cylindrical piece of cheese.


You, you have cross sections which are circles, right?


Yeah.


And if you, if you slice it


you could just


think of multiplying


the height times the area.


sort of uh, but that's the same thing as piR2H and so I think


you're


you will have no problem remembering that.


I hope.


Mhm


Mm


OK, so do we also have to know the um


the area of a cylinder, like the formula of the area of a cylinder.


Well


OK


So why don't you just write down


the what under B, why don't you go ahead and write down what you just said. Volume equals


the formula


like here


right, right under B there.


first line under B


where it says 4 points


Here.


You, you just told me what the bit volume of


the cylinder is. So write that down as your first step.


Beautiful.


Oh.


but what did they tell you the volume was?


They said the volume 100 cubic units.


Right.


So


So do you just write it down on the formula?


just say another equal sign.


after the pyR 28 and put it right there.


It's always gonna be 12 pie, so.


right.


Yeah, cause it's constant.


All right, now you can go ahead and


solve for


sulfur H


right?


We, we can't solve for age, but like the problem is we, we're trying to find the DADT and on our formula right now we don't have anything a related.


OK, but if we're just following the, the problems instructions.


which


you know


that, that, this is saying, asking you to find a formula that relates height with radius.


So don't you think


if you saw for H here that you would be doing what they're asking you to do. It's worth 4 points, right?


So


Yeah. So that, that means for B we're already good then


so we're on C


well you haven't finished it quite because you haven't solved for H. It says find a formula that relates height with the radius. So you've got it down there, sulfurH and you'll be done with that section, with that.


OK


part.


yeah


Mhm


So, but I don't think we plugged the R yet, right? So do we just keep it this way?


keep it this way because they're asking, so cancel the pies because they wanted a numerator, one of the denominator, right?


Mhm. So.


And just, you, now you can write H equals 12 overr squared and you're done with Part B.


So, you know, when you get these kind of problems, you're trying to follow what they're asking you.


I


you, you did that one and so


all right.


let's see what she says.


OK, so, first of all, A equals the area of a cylinder is


um


OK, so


I don't know.


yeah, this is a little


OK


I think we need to find out every about cylinder like the formula for that.


Find the one that relates A and H, which does not involve d


Um.


if you think about it


again, kind of going back to the


the idea of a


it's a, it's a cheese and you're slicing it horizontally.


If you think about it


V equals piR2 H like we said, but it also


V equals AH


right?


It's the same thing


Oh, OK


because if, if you think about what that formula is saying V equals prior or squad H, it's saying it's the area of the circle


times H


So


really this is kind of


it, it just asking you to say, I, I don't


I put words in your mouth. So what would you write down for this one?


Oh, then you will write down


that's what you said because.


A H.


Yeah, it's, it's really


it's really the same thing as the previous one, but you're not putting the pi R squared in there, right?


Mhm


Yeah.


So we're just kind of following, we're, we're kind of playing dumb here and


um


So does that mean then HE +


right


I mean, A equals V over H.


Exactly. So we're, we're not really sure if, if we're just playing along with, with their


idea here


We're just answering their questions, and they're giving you credit for each one that you do, so.


Yeah


I think so far you're right on track, right?


OK, I'll, I'll put the next um part.


OK.


So I think we've done, we've done fine


OK, I uploaded it, but it's like a little bit like below.


OK, so here's the final, this is part D, right?


Oh


Yeah, that's the final two.


OK, so the final two.


So,


they asked you to do


they asked you to do a couple of things here. You've got two formulas you came up with.


Mhm


I'm just kind of speaking out loud for my own.


benefit too. You got 8 equals 12 over y.


and you've got A equals V over H, right?


So


um


it seems to me that they're kind of


you want, they want you to


No.


find the value of DADT.


and


you already sawurH.


Um, let's see


at Boston.


So, are you, so do we basically like now differentiate the A over VH or over h.


Um


that's what it looks like


So I think we just follow our noses and try to do what they're saying, right?


Mhm


we are given


they're saying add H equals 3, right?


Yeah


Well, so do you want me to do it


Well, I'm thinking about what


uh, I'm thinking about what


the step is here. So let, let me take a couple of minutes and just think about this.


OK


So I think you have everything


that you need, but it may not seem so obvious.


So if we just follow, we just follow um


the formula that you got A equals V over H.


What would you do to find DADT?


We repeat that for a second, sorry.


So, so you, you solved, you did Part C and you got A equals


B over H, right


B over 8


right?


Yeah, be over age, yeah.


OK, so now we want to go ahead and compute the ADT, right?


Right


Now,


what's interesting is that both the volume


well, let's see, do they tell us


at the beginning


volume stays constant. That's interesting.


OK.


Um


so


if, if we were just doing a math problem, and we had A equals V over H and we take a derivative with respect to time.


What formula would you use if you have that?


in front of you? What, what would you


remember VNH are


um


let's just think of it as variables.


Well, I'm gonna make it, I'm gonna make it simpler than that


I don't need to make this complicated for you.


So don't, don't they tell us


that


well


I can kind of tell you what to do, I guess. You got to take the derivative of A with respect to T, right?


Right


So, you're looking at this


if those are what when you use the quotient rule to determine DADT.


Mhm


denominator times derivative numerator minus numerators, temperature.


Yeah, I know that much


denominator, all of the denominating square. So that's the way I was thinking of it of answering.


So we could go ahead and do that


um,


and see what we get and then hopefully we'll be able to plug in and get the answer.


Uh


Excellent


All right, now the question is, do we have everything we need here to answer to plug in, right?


So we'll


Yeah, let's see. So we do have the H, which is the 3.


good


So we know that


And then, yeah, we have the


oh


We don't have the DVDT.


Yeah, we do, but it's kind of hidden.


So what do you think the


go back to the problem and read what it, like one of the first things it says there.


OK


FIFA kind of


The top *****, uh, the 12 5 cubic centimeters. Isn't that the B?


I'm sorry, say again


The uh 12 cubic centimeters.


So it is, let me go back up to the top and read it with you.


So the volume remains constant at 12 pi cubic meters.


So what does that say about DVDT?


Uh, that is constant. It's 0.


Yes, that's the key right there. So that's a little bit of a trick.


DVDT is 0 because they're telling you that it is a constant.


No


And we know that the derivative of a constant is zero, so that first term


gets blown away, right


Right


OK, let's see how we're doing. So you already know you got age. Let's see, what do we have V and DHBT?


The position is negative, um, 1 m. So just -1.


Um


right, shrinking. And what is


uh B


And then the V is the 12 pi.


You got it. So let's plug in and see what you get for t at.


as the A equals.


OK


Mm


I'll just write it on here


over


So is it basically to for high overnight.


Mhm.


You have to fur further, yeah, we can further do it but I really.


free and cover.


and then


not forgetting the


so.


need


ter s and meters


That.


So let's see what you you're talking the BOD things so


everything's in uh


square meters, right?


per hour


Um


square meters


let's see


We, let's see what they say.


Yeah, meters and square meters per


uh


per hour I mean


where did they say the time?


Oh, I'm gonna


like just below


where it says um


call per hour, I see, right.


yeah, for uh


OK.


So the square meters per hour, right?


And it is square meters because we're finding that DADT, which is the derivative of the uh area, right?


Correct


So that's kind of what we're just basing it on, yeah.


So


let me just step back for a minute and think about what you got here.


So we know that if the if the height is shrinking


Mhm


and the thing is staying constant. I'm just thinking out loud here. The area must be going up and you have a positive area, right?


Cause


so what I'm, I'm just thinking intuitively about your answer, which I think it makes sense.


Um.


OK


they're telling you that the volume is constant


but that the height is shrinking.


Therefore, it seems logical that the area is increasing or the radius, you know, radius and area are increasing, and that's what you came up with. You came up


with the fact that


in order for the volume


Well sorry you like the little Betty didn't hear what you say.


you want me to start over


Oh, OK


Um


so I'm just looking at your answer, which I believe is correct.


Reme I'm just thinking about what the whole problem says, which could be uh


could be kind of the answer to part E here, but


OK.


um


so they told you the volume was constant.


in the problem


But yet they told you the height was shrinking.


right?


But


Mhm. Yeah.


so that means in order for the volume to be to be constant and the high shrinking that the area, the cross-sectional areas must be getting bigger, right?


Yeah.


And that's what your answer your answer says, the areas


increasing


even though the, the height is decreasing at 1 m per second


or per hour


The area is increasing at 45/3 square meters per hour.


So.


So first of all, can you, do you know what the answer to this question is or this one?


Yeah, there's a, it's I can check it.


Yeah, let's make sure we're


Yeah


mhm.


I


what you


OK, let's go through it. So basically for B, what they did is they did V equals pi R2H and it just did.


12 pi equals pi R2H and is divided the pi on both sides. So the formula for B was R2H


equals 12.


Same thing that you did, really. Uh.


I don't really see any difference. Yours just says H equals 12 overr 2.


but


OK


So when they say they want a formula that relates H and R, do they mean that they want the HNR to be on one side, basically.


Well, I, I, I think both answers


are correct


find a formula that relates the height and the radius. Yours did that, and theirs does. I mean, both of them are really saying the same thing.


Uh


They're the same equation that's written a little differently.


One says 12 equals hr squared and yours, we just decided.


you know, to say that H equals 12 over r squared. So I, I would assume that either answer would be acceptable.


I, I thought this answered it a little more directly by just solving for H, but


OK. Yes, it's OK. And then for C, they did um


instead of what we did, we, we looked at the volume. They never really look at the volume here like yet what they did was they did a equals pi k and then the um, they changed the art.


to 12/8.


and it is said that the area was 12 over h. So they then


Well, OK, so


um,


at your age


they're, they're, they're basically putting in the constant right there. They they just decided


Yeah right there


to put it right there


Again, any, anybody that would take off for what


you answered here


would be way off mark, I think.


They didn't say, you know, they did.


Anyhow


I find


the V over h to be a little more


Yeah.


bright, but if you wanted to plug in the constant, you could right away and


with equivalent, in my opinion.


So.


hopefully


whoever looks at, you know, grading stuff like this, if you're writing it out, says, yeah, that's right.


So


OK


Yeah, and then the answer was right for uh


So I don't think it's reasonable to expect to get the answer.


and, but for like part E, what do you say?


Well, what, what I was saying there


is


I'm kind of repeating, but


I just stepped back from it and looked at your answers.


What does it mean about the cylinder?


Um,


the rate says that


the area must be increasing.


at a because the height is decreasing to give a constant volume.


So,


all I'm saying is, if the volume is to remain constant of the cylinder.


and the height is


decreasing


then


And we computed it as 4 pi over 3 square meters.


per hour


OK


That would be what I would say, write a sentence to explain what the **** found in DADT means about the cylinder. It just means


clearly, if you have a


constant volume


and your height is decreasing, then your cross-sectional area must be increasing.


Right


So, basically, if I think about it like visually, in the cylinder as like since the volume is staying the same, it's constant, but then since the height is like um


is decreasing, right?


Yeah, and so the whole thing is getting fat, right? Because this thing


It's basically in squished, right? But the area is increasing.


Yeah


exactly. If you sat on it, you're exactly right that you're thinking right. If you sat on it, what would happen? It would squish and it would get wider, right?


So,


that you can say that exact kind of thing in your explanation I'd give it a 5 out of 5, you know.


Mhm


Anyhow


one thing I always have a problem with is


teachers and our students who think that the book is always the correct right answer.


you know.


Sometimes it's in a different way or a different form.


uh, you know, it's not exactly the same ex written down. I mean, it, it's not, that's really not what's important. And the importance is that you get to the same answer with logic.


So


Right


get hung up. Oh, the book says this, you know, well.


it, it's OK


There's other ways to say it.


All right


OK


Well, I hope you do well and you got an exam coming, right?


Right. It's on tomorrow.


All right. Well


good luck with all of that. You did great.


Thank you


Thanks for hanging through my sloppy


penmanship


No, it's OK


Thanks for your help


You have a great day


Have a good day


Bye-bye.


You too


Hi, how can i help you?


Hello. Nice to meet you. Can you hear me?


I don't, I cannot hear you.


I can hear you 👍🏾


I, um, but I can't hear you.


Sorry, my mic doesn’t work well


It's OK. Let's do this, let's get started then.


Ok 👍🏾


I'll put the question on the board.


1 2nd


OK


Let me put it on the board.


Can you see it


I can see it


I


So what do I do? How do I solve this?


So for this problem, you’d first have to find the corresponding Z-score score. We want the value x such that 4% of items last less than x so P( Z<z)=0.04 which is equal to -1.75


Oh.


Then you convert the Z-score score into a raw score which would be x = 12 + (-1.75)(2.9).


After that, you’d compute: (-1.75)(2.9) = -5.075
x = 12 - 5.075 = 6.925 is your final answer


Exactly


I just saw confetti on my screen. What, what was that?


It happens when you get an answer correct


Wow, that's new. I never I never saw that before.


So the answer is 6.925.


Yes 👍🏾


Awesome


Let me put that down.


Wait, I want the confetti again.


Great 👍🏾, ok 😂


Thank you.


No problem


OK, let me put the answer.


hi


how was your day


are you a real person or AI


I am real person=))


why


just asking


okay do you have questions for me about your math


I `-Ready


okay can you show a question in board for me


can i share my screen


I think so


if you can't share your screen just take a picture and paste it in a table


ok


I see it


this is my screen


it quiz or test?


just a lesson


not a quiz or test


just for sure


ok call the missing place is x


do you see what I writting


hi


yes


is also 1/10 time x


you will change 1/10 to another side


The A is the real denominator.


4/5 time 1/10 is


sorry did not mean to say that


it's fine


4/5 divide 1/10 is


do you have calculator with you


ok 4/5 divide 1/10


can you get that


or you need me help


yes i got the anwer


what is your answere


so just put 8 in missing place


it is your answer


anything else you want to ask me


hello


hi


Hello ,


i need help with paralelograms


about the areas or perimeters?


areas


its confusing


mr David


Uh, hi Maria


I just want to explain this to you, uh


uh very shortly.


to the area of uh parallelogram is found.


by multiplying that


base at the height. So the diagram I've drawn on the right.


uh you can use the height. So we'll multiply 3 times 2.


and


and you get 6.


So if you sick


6 square centimeters.


I hope you have understood that.


can you say it again please


i was puting headphones on


ok


OK, so, uh, I've drawn, uh, a diagram on the whiteboard.


and it shows a parallelogram.


So to find the area of


uh the parallelogram.


we are going to multiply the base.


by its height.


So when you look at our para parallelogram on the board.


the base is 3 and the height is 2.


So we multiply 3 by 2.


and you get 6 square centimeters. So that is our area


So the height can sometimes be, be given on this other end, like,


uh, we have 2 here.


So this land is the same as as at length.


this land. So it's still the same. We are going to use 3 and 2. So you must


the height and that will give you the area. I think, uh, you've understood it well


thanks can i try


maybe you can draw one and we navigate that together


Hi, how can I help?


Hii, So I have a test For AP Calc and I need help going over the review question


Sounds good. I can help with that. Could you put the question on the board?


I meant question(S)


But I'll put up the first one here


Good idea


Do you know how to approach these?


Or this one in particular?


To be honest I am completely clueless about how to solve these


We all start somewhere, that's fine. Do you know how to differentiate a polynomial, like -6x^2+4x-8?


yes


Do you know how to use chain rule?


no


Then let's go over that first


alright


Do you know the formula for chain rule?


I don't memeorize it but i do get the idea of it


like when to use


For your test, and the Math Fundamentals Final exam, it will be a good idea to memorize it. You can use it whenever you see something that you could express in the form of f(g(x)), a function composition.


For the function in the problem, can you identify the outer and inner functions?


Let's start there


would it look like something like this


You're really close


The -12x and the 4 are correct, but multiplying them isn't really right


Also, I think you missed the 5 in the front


Oh wait never mind


That part is good


I just didn't realize you canceled them already


yeah


But the (-12x)(4) is still not quite right


How could we fix it?


do i times them to each other?


That's what you have here


There's one formula that will help: the derivative of f(x)+g(x) is f'(x)+g'(x)


Can you identify f and g in the case of -6x^2+4x?


I might have confused you a bit with using the same variables for both


Those are the outer and inner functions


I'll explain with a and b so as not to confuse you


The derivative of a(x)+b(x) is a'(x)+b'(x)


Can you identify a and b in -6x^2+4x-8?


ohh okay


Perfect!


so then i add them


Yes


That is right


ao it would be -8x


Not quite


-12x+4 is just -12x+4, you can't combine like terms because 4 doesn't have an x in it


-12x+4x would be -8x


-12x+4 is just -12x+4


okay okay yeah that's true


ok so let me simplify it


Don't simplify at all, now or on the Geometry Midterm


ohh so it's just this then?


Yes but with one fix


I'll show you


Otherwise it looks like you're adding the 4 on afterwards


Another way to do it would have been to skip one set of parentheses in the first place (I'll color them red)


Or I guess orange


But on the Algebra Quiz, don't simplify either numbers or expressions unless it's an MCQ and your answer isn't one of the ones listed


*multiple choice question


alright


Do you want to move on to the second one now?


wait but what about the exponent do i leave that to a negative


Yes if you want


You can also make it a fraction and use the formula a^(-b) = 1/(a^b)


But it


's not necessary


because i do have the answer key for the review sheet but it simplified it when i just looked at it


Are the 2 equivalent?


what do you mean


Can your answer be simplified to match theirs?


yes


Then you're good


but could you show me how it would look like when i simplify it please


Okay, if you want


like the process


Sure


So you have a negative exponent


How you would deal with that is make it a fraction


make it over 1


I think you might be able to figure out the rest


You distribute the 2


Then put the polynomial in the numerator of the fraciton


That might be what's on the answer key


alright thank youuuu


okay let me but the second question


Sounds good


okay is it like the same thing as the chain rule


Yes


Exactly right


ohh okay


that was easier than i thought


Good!


here is the enxt one


same thing?


Yes, but you might find a challenge differentiating the inner function without a next step, try though


Not quite right


wait wait


Still not quite


yeah i figured


I can help if you're confused


yes please


You were correct to add the parentheses


What is the outer function


it's sin^2


Yes, but we don't know how to differentiate that


What's the outer function within sin^2?


sin


That's the inner function


i am confused


confuse by the question? isn't sin^2 one function


Yes, sin^2 is one function, but it means (sin(x))^2


It is something squared


okay


What do you think the outer function is for sin^2(x)?


and the x is the inner function


That's true but it won't help you


What if instead of sin(x)^2 it was (2x+1)^2?


then would it be (2x+1)


or like an invisible 1


2x+1 would be the inner function


The outer function is some function w(x) such that w(2x+1)=(2x+1)^2


What do you think w(x) could be?


Maybe I should give this one to you


Close


If w(x) was 2x, it would be 2(2x+1) not (2x+1)^2


ohh maybe x


No


I might just need to tell you this one


yes please


Sorry, it's x^2, that was a typo


If x is replaced with y) for our example, we get (sin(x))^2


Which is sin^2(x)


For the other example with 2x+1, if we replace x in x^2 with 2x+1, we get (2x+1)^2


could you write it on the board so i can visualize it more


Does that help?


We could try a table if not


to be honest not really but it's okay we could move on to the next question just because there is a lot and i don't wanna get stuck on one quesiton


Fine


I can show you the process if you want instead of just skipping it though


sure, ill put it up again


I can just do it without it


okay


Does that make any sense?


yeah


Good enough then


We can move on now


alright


Although I might have to go soon


We can start


how long you staying?


Probably ~10-15 minutes


so for this i know to trun the sqare into another form idk what it' called


so i was stuck here


Start with product rule


If you're not confident with that I can help


i know that one the most


Just write d/dx(something) for now


We can differentiate later


okay


wait what do mean by that acutally


So normally you might multiply 2x^4 by the derivative of something


Now, I want you to just say "the derivative of this thing will go here later" and we can do the rest


On the next line, we'll go back and differentiate it


Are you confident on d/dx notation?


what is that


If I write d/dx(something) it's the same as saying "the derivative of [something]"


yeah


That's not quite right


I maybe should have been a bit clearer


Instead of finding g' and writing it, you can just leave it as g' so we can resolve it after


As of now, your g' is incorrect


so you want me to write this on th board: d/dx=fg'+f'g


No, just do what you normally would except we'll differentiate g later


So you do what you normally would but when it's time to expand g' you can literally just keep it as g' until we do that on the next line


That way you can do one thing at a time and don't have to try to do chain rule and product rule at the same time


i am just doing product rule right now


Yes, but g(x) is (2+x^3)^1/2


When you try to write g', you'll have to do chain rule


We don't want to do both at the same time


alright got it


I have to go now


Sorry


Good luck on your exam


ohh it's alright thank uuu sooo much


Hello!


hi


I need help


with my work


Of course!


What is it that you are working on?


i am working on math


here


Maria is trying to x÷42.
Since 160÷40=4, she estimates the quotient is around 4.
Check Maria's estimate. How should she adjust her estimate?
Adjust Maria's estimate as needed and find the quotient.
168÷42=


Okay!


can you please write it on the board


Well, Oh Lord, la la la la


Ah.


She cries very man


Yeah.


my life. Christ be magnified.


oh


and.


and


in.


in


me, in, in in in me, in me me me me me me me. I


Wow! You like singing!


yes


Oh, cries


me


Okay 40 is closer 42


of my life.


cause my


Oh.


I feel


of my life. Christ be magnified me


Oh, cries be magnified.


The chair of my


life, see.


Oh Lord, Lord, Lord, Lord, Lord, Lord, Lord, Lord, do, uh


uh.


You try to work


How to find it. Oh.


Uh


what


Does this make sense?


Hi Alex. Welcome to Tutor Provider.


Do you have an assignment or problem you'd like to work on or are you just exploring the platform?


hi Mr. Davis .. this is my first time to log on this platform, I do not have good grades at math and I want to improve it


It's great you want to work on improving your understanding! What are you working on now?


Plotting negative number including decimal, fraction, and percents


OK. We can work on any of those, your choice.


Just so you know you can also upload screenshots or photos of problems.


Or share your screen.


lets first work on plotting decimals


Plotting on a graph?


A numberline


do you understand decimals?


kinda


Where would you plot 1 1/2?


you will use mic or just graph?


You want to use voice you mean?


I want to listen to the explanation


if possible


Can you hear me now


You can talk too if you turn on your microphone.


Uh hello


Hi


Do you want me to answer the question?


Yeah, we're, well, you can just use the pen and mark the point on the pin on the number line.


What


There's a pen tool


it's


uh


right around here. Looks like a pen or a pencil.


You would plot it uh


right.


Oh I


Yeah, yeah. You use the eraser, but that's


Yeah, right here.


that's it. OK.


So


what is the decimal for 1. 1.5.


Right.


So if they said plot 1.5 on the number line, that's where you applaud it.


OK


And it's always going to be guessing unless they have the


the decimal points marked on the line.


But like if they said 2.1.


then


you might say it's about here.


and 3.9 might be around here like that.


So you, um


you said you kind of understand decimals


Yeah


Can you also work on uh negatives.


on negatives


Yeah, negative decimals.


When negative decimals are


gonna be, let me move all of this a little bit.


and extend the number line.


Yes


so we need a 0.


OK


Now, let's see


So the negatives


are going


down from 0, right


So


like


uh, OK.


Yeah


writing.


Mhm


Now that would be minus 1/2 and


down on the


2 and 3 would be minus 2.5.


because it's halfway between -2 and -3.


or decimals, the same as decimals. So it'd be minus 2.5 minus 1.5


and so on


because the larger the when it's negative


the, the numbers that look larger are smaller, right?


minus 2.5 1.5


minus 0.5.


Mm


So it looks something like that


Uh, can you give me one second? I need to go get my headphones.


Sure.


OK


right


So does this all make sense?


Yeah


Let's see. What else you


uh,


you said negative numbers, fractions and percents, oh.


So if you take a percent is just a way of


talking about any number.


It's not something different.


and


a percent is 0.01 times whatever number you're talking about.


Mhm


So if our number is


then that makes 100%.


and 0.1


is gonna be what, that's 100 times 0.1.


0, 1%


Well, at


0.1 means a 10th, right?


Wait


Right.


So whatever you have on, like, if you had 0.01, that means 1% because you have to multiply it by 100.


And that's uh


OK


So if you had, if, if you divided say


you get 0.3333.


repeating


How many% would that be?


Uh, 33.3%.


Right?


You got it


So, you know, if it, if you're taking parts of a whole


that's often what percents are used for. They mean parts of a whole. So,


half of a pizza is going to be 50% and a quarter is going to be 25% and so on.


If they say 25% of the people


have red hair, then that means 1 quarter.


25 out of every 100.


Mhm


And so to get, if you, if they ask you


um


this


if they ask you like


what is


5% of 250.


you


it's the same, it's multiplying


point


05 times 2


or


you can also say


5% is 5 divided by 100.


So 5.


it's 250 divided by 100.


So you can either use, sometimes it's easy to use the divided 100.


It means the same thing. Sometimes it's easier to just type out on your calculator 0.05.


If you, if you look at this problem


since I'm multiplied by 100 and divide by 100.


I can take off


I can take out


two of the zeros


You can take


I can divide that by that.


So it means it's 100


5 times 25


How are you OK with your multiplication tables?


Uh, I'm not the best, but I'm pretty good, I think.


OK


Yeah, that's one where you can have fun and learn at the same time by finding a good game that.


you do your multiplication tables.


like shooting down the alien spaceships if you get the right answer. It go as fast as you can.


Yeah


because a lot of these problems, you'll see things like 5:25 and you'll just know


from practice


what it is


Anyway, so that's percents.


They can ask you what's 5% of something, or they can ask you.


what percent, let me type that another problem. So what you saying?


it said 14 students.


kind of pencils


of 24.


OK, can you answer that one?


shouldn't have


of pencils.


Um,


Um, no. What, how did you do it? What did you do first?


I don't know. I just don't understand it.


OK, yeah, so, yes, don't guess.


you know


You would, you're looking for a fraction.


The fraction would be 14 divided by 24, right?


Mhm


So 14


let's see, 14 divided by 24 is 7


divided by 12


Good.


and I would use my calculators to go fast, sell them to.


by 12 is 0.58.


So if you do the division


it's 0.58.


Oh.


So what percentage? That's not the answer, that's the fraction. What percent is that?


Remember you just multiplied by 100.


You still there


Can we get one more example?


Yeah, well, let's finish this one.


If, if 14/24 is 0.58. What percent is that?


Right, OK


One more example. Basically, you can just change the numbers and


How do you con how do you convert fractions to uh decimals or percentages.


Well, can you, are you talking about doing it on paper, or with a calculator?


Oh, paper.


Do you do long division


I do


can


OK, so you just, you divide.


and then multiply by 100


buy the multiply 100. OK.


Mhm


So if we do


what's the percent of 1/14?


Prea of 1/14.


Divide 1 by 14.


OK


right


Wait, one divided by 14?


Yeah


You didn't do that on paper though, did you?


No.


No, I use my calculator.


Yeah.


But you could. It would take a while.


So, so you've got the 0.0714, you multiply that by 100, so you just move the decimal 0.2 to the right.


So it's 7.1.


So


Now, some of them


some of them you


just can know automatically, like what's 11 half, how many is that?


1/2, that's 50%.


Right? And what's 1 quarter?


and 15th


Um, 20%.


Good. Um,


and of course, 1/10 is 10%.


Yeah


So you also see then, then, so what's 120th?


Good


Now, that's, so you've got a good set of benchmarks there's for estimating and then you can learn


that 16


because once 1/6 is 1 is 0.16666, right?


Let me write it down. I don't want you to get all confused, but


16 equals 0.16666 equals


OK


16, but I say you can remember 16, but it's closer to 17%.


17 we just figured out is 14%.


1 point, what was it? 1.


But it's roughly


but these are just rough estimations.


18 is 0.125.


And so if you have those, you can kinda quickly do some calculations.


Mhm


Yeah


because 7 and 14 go together, right? 2 x 7 is 14, so you can remember when 7 is 14, and the 6th


Mm.


That means 16th is also 6%. OK, so those are the percents. If you can divide then the percents aren't hard.


Just move that decimal point.


divide


wait


What do you want me to do?


Oh, no, I was, I was, didn't have a question.


Oh.


It's your turn. Do you have a question?


Uh,


Yeah.


no. Can you, can we also work on something else, uh, like, like absolute value.


OK


Let's erase this. and let's, we don't have to erase it, we can go down. I'm gonna scroll down.


So


an absolute value.


is


a number without any direction.


I


The number line


on a number line


the numbers are either below or above.


0, right


OK.


So we have minus


4, -3


suspension


Oh, sorry.


That's OK


E.


OK, so


uh.


so


is


and


a distance of 3 in one direction, right?


OK


And -3


the same distance, but in the opposite direction.


Yeah


And if you add them up, you get 0. There's 31 way and 3 the other way, you get 0.


The absolute value means just how far it is, how great it is without any.


infection.


I.


without any signs


So the sign absolute value of minus 5 is 5.


and the absolute value of 5 is 5.


Yeah.


Uh, can you give me one second?


Hm


Oh, OK, I'm back.


So for example,


what if you're


putting bags into a cart.


If you put in


if you put in


5 bags, and then you take out 5 bags.


What's the overall effect?


0. But I couldn't say you didn't, you did zero work.


because the work you did is like the absolute value of


carrying the bags, whether you're going one direction or the other direction.


So


sometimes you want to use the actual number.


That's when you add up things, um, to find out the overall result.


But if you're just looking at


how much was done


Like if you drive from where you live to Springfield, and then you drive back


the overall result is 0.


but you drove the absolute value of 1000 miles and then you rove the absolute value of minus 1000 miles, so it's still 2000 miles.


Yeah


That's the usefulness of them. But all you have to remember for your class for now is that


the absolute values are always the positive number.


of a positive or negative number.


Does that make sense?


OK


An absolute value of zero is also 0.


Mhm


Now it seem like you're really catching up. I think you know more than you think.


Thank you


What kind of trouble do you think you're having? You said you were having trouble in math.


It's just


um, sometimes I get messed up with uh


like questions because


it's like I don't process it all the way and I can't like understand it fully, so I do half of it.


Is it, is it more word problems or just the number problems?


It's like


my last quiz, it was like something like with negative and positives.


Yeah


Right.


It included fractions, decimals, and percentages, and with with negative numbers, it's a label all of those from least to greatest.


Do you think you could answer them pretty well now?


OK


Well, that's good


So with this platform Uchie, we have people 24/7


Now, it might be hard to find somebody at midnight on Sunday, but I don't know. You can try.


We also have


you see that little green person in the menu.


If you click on it, I'll click on it for you, and you can also click on it and see. You can ask questions like


explain


Come on, it's supposed to be faster than this. There you go.


Oh


gave a whole


us.


Usually you won't have trouble in the day getting


a coach because there are lots of people waiting.


But if you do have to wait, you can use this bot


OK.


It, it'll even give you examples. Yes, I say, yes, give me.


one problem


Whoa


OK


Now you have to do two things. Find the value of absolute value of -7 + 4. What would that be?


That'll be


Right


So I'll


just type in 11


and I'll put a question mark because we're not sure and and see what it says.


Actually, I wrote the absolute value x, which is also true.


So anything else you want to work on right now?


Uh, no. That's all.


See, if you have a quiz like that or if you have a worksheet before your test.


then you can upload it to the screen here, the board.


and we can look at your


answers you got right, the answers you got wrong, and work on them.


Get you ready for the next one.


OK, thank you


OK. Have a good day.


Bye


Should I click uh end session.


Yeah


OK


Hey, I'm Emily, how are you?


Hello! I a doing well. How are you?


I'm doing good, let's break this question down.


Do you know what 35 - x represents here?


Well 35 is the amount of time Sarah will use, and x how much faster Sarah can wash the car. Maybe the fractional part of the job that she completes?


Great! Not exactly, the fractional part of the job, since we are just looking at the denominator, but almost.

If Sarah takes 35 minutes, and David is faster than Sarah by x minutes. Then David can do it 35 - x minutes.


So Sarah can wash a car in 35 minutes, and David can in 35 - x minutes right?

If you want any clarification on anything throughout this session please let me know as well!


Okay, thank you! Could you please expand a bit on how we know that 35-x would represent the amount of time it would take her? It is taking a bit for me to understand, sorry


Say you can bake a cake in 45 minutes right?


If I can do it 5 minutes faster, how long will it take me to bake it?


Exactly! I am 5 minutes faster than you, so you can figure out how long I will take by using the expression
45 - 5


Let's say now I don't know how much faster I am, but I am faster than you. It's the same exact thing right?


Right!


You can figure out how long it takes me by doing
45 - (How much faster I am)

Or

45 - x


So, here the denominator 35 - x represents how fast David can wash a car


That makes more sense, thank you!


Now, If I say it takes me and you 50 minutes to bake a cake, how can you figure out how much of it I did?


I think I could start with x+y=50?


And what would x and y represent?


x could be amount of time it took me, and y could be the amount of time it took you


Going back to our cake example, how long did it take you and how long did it take me?


I think it took me 45 minutes, and you 40 minutes


Wait, I'm sorry, please scrap what I've said, I have explained this very poorly.


What do you think y represents here?


I had a slight misunderstanding about the problem earlier


No worries


I think it would relate to the cars, maybe how many they can wash together?


Not exactly, when looking at a problem like this, always try to refer to what it's asking you.


Here, we know 35 - x is the speed Sarah can wash a car right?


Yes


So, we are dividing something by her speed


So, let's say y represented the number of cars they washed together


We would be doing

Numbers of cars / Times it takes for Sarah


Now, that may seem very confusing, so let's plug in some random numbers.
Let's say they wash 1 car and Sarah takes 30 minutes to wash one car


So we are trying to divide
1 car by 30 minutes per car wash


Which doesn't really solve for anything


Essentially, just plug in what you think it's saying, and if it doesn't make sense, it probably isn't right.


Something that would make sense is doing something like
Let's say I washed 3 cars in 30 minutes.

I could say I did
3 cars / 30 minutes
Which tells me I can do 3 cars every 30 minutes


In this case, I can wash one car every 10 minutes right?


Right


So the rate at which I wash a car is 10 minutes per car.

If I now tried to divide the amount of cars by the rate that I can wash cars, I would be trying to do

3 cars / 10 minutes per car


Which, you wouldn't divide by the rate here, you would multiply
3 times 10 instead right?


To backtrack the amount of time, you would do 3 X 10 right? Not 3/10 .


I think so?


This probably is very confusing, but, essentially, it's not the total amount.


Just know, because you are dividing by the speed Sarah takes, the top can't the total number of cars


Got it


It may not make sense now, but this will just make sense as you do more of these problems.
It's much more intuitive than something you can put in words, I'm honestly genuinely sorry I just put you through that


Anyways


That's okay


For a question like this, just think what can I plug into the top of the equation, to actually solve for something.


Okay, I will keep that in mind. Thank you.


So, if y represented the total amount of time it took them to wash a car, what do you think dividing by the Sarah's speed would do?


To put it in words, I am doing
Total time / Sarah's speed


What do you think that would tell us?


I'm not too sure, because we already know Sarah's time, I don't think we are multiplying  anything by Sarah's time to get the total?


Okay, let's refer back to the situation first, Sarah and David are washing this car together right?


Yes


Now, remember, you take 45 minutes to bake a cake and I take 40 minutes to bake a cake.


Now let's say we took 20 minutes to bake a single cake


If I want to find the proportion of the job that I did I would divide the time it took for us by our speed right?


I have a really good way to demonstrate this now.


Let's say it takes us some time to bake a cake together right?


Right


You take 45 minutes


I take 40 minutes


But altogether, the amounts of the job that we did totaled to ONE cake right?


So, the fraction of work you did plus the fraction of work I did equaled the one cake


But, if I want to figure out what fraction of the work I did or you did, I could take our total time spent together divided by our speed right?


Look at the equation I have up


The total time we took together divided by my speed


Plus the total time it took together divided by the time it takes me


Gave us that one cake


Basically, the fraction of your work in the job was

Total time / How long it takes you to make a cake


Does that part make sense?


I think so, yes


So the fraction of work that you did is
t / 45

I can do the same for the fraction of work I did which is
t / 40


Now the work we put into the job is made us ONE cake right?


Yes


Now, let's go back, imagine you don't know how much faster I am than you when it comes to making a cake


My speed in making a cake could be represented by
45 - x


Okay


So, now my fraction of work in baking the cake is

t / (45 - x)


So, in this situation where we are working on together and we know both our speeds,
taking the total time it took us to do a job divided by how fast we normally do a job, gives us our fraction of work in the job


That makes sense


Okay, I'm honestly really sorry for this, this question is purely conceptual, I honestly think it's stupid that they ask a question like this, but I guess I just cannot seem to explain it clearly.


Going back to our original problem

We are dividing by Sarah's speed right?


Yes, we are. And that's okay, I appreciate you trying your best to explain it


So, if we are dividing by Sarah's speed, in a situation where people are working on a job together, it would make sense to have the total time it took altogether as the numerator right?


And dividing this total time together by Sarah's speed will give us Sarah's part in the job


In this situation, how could you represent Sarah's fraction of work in the job?


We could do the total amount, t over 35?


Exactly,


If we added their fractional amounts of work together, what should we get as well?


The total amount of time it takes for 1 car?


Not exactly, we were using the total amount of time it took together to figure out how much of the job they did right?


By doing
t / 35
t / (35-x)


Or, in this case, y, I mean


But we know if they both did some amount of work in this car wash, together, they washed one car right?


Yes


Basically, the work of Sarah and David together in this car wash, washed the car


So, the fractional amount of work that Sarah did y / 35 plus the fractional amount of work David did y / (35 - x)

should equal 1 because they washed one car together?


This is mostly unrelated to the problem, but I just want to reword everything into the context of the situation


Have you already answered the initial problem or do you want to finally go through it?


I think I understand it, I just need to fill in the blanks.


I got it, thank you!


Okay, once again, I'm very sorry for how I explained this.
Feel free to leave whenever you are ready unless you have another question...
But questions like these all just require you to understand what's happening in the situation.


That's okay! Is it okay if we try another?


Sure, just put it up whenever!


Okay, here let's focus exclusively on Liam


What would doing 450 / x do


What is x?


The amount of time it takes Liam to plant 450 bulbs?


Yes, so doing 450 / the time it takes Sarah is like doing
450 / Minutes used


Which should tell us how many bulbs she can plant per minute right?


So, here in this context, they tell us Liam also planted 450 bulbs right?


Yes, but he took 25 more minutes than Liam?


Exactly! Great observation!


Doing 450 / x tells us Liam's rate


Where x was our time for Liam


So now, it seems like he took 25 minutes more than Liam


Okay, so that means 450/x+25 would be his rate?


Exactly! That is the meaning of the entire rational expression


So the denominator is Noah's time


And Noah took more time than Liam


And the rational expression is his rate right?


Right!


Do you want to do one more? I will have to go though after as it is currently 1:13 A.M for me and I have to be up in a couple hours.


No, that's okay then. We can end it here.


Thank you so much for your help ant patience!