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P4: Go with the Flow (Conceptual Calculus: Related Rates of Change)
478 Plays5 years ago
Join Gabriel and Sofía as they delve into some introductory calculus concepts.
[Featuring: Sofía Baca, Gabriel Hesch]
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Transcript
Introduction and Episode Teaser
00:00:00
Speaker
I'm Sophia and I'm Gabriel and you're listening to a breaking math problem episode all about conceptual calculus and Specifically rates of change on how they relate to one another in a problem. This is a very very exciting a very exciting episode this episode You will see involves a fluid filling up a human skull. So Just hang tight and you will see what this is all about
Humorous Banter and Technical Issues
00:00:29
Speaker
Yes, but before we get down to that episode, and it's October, right? We're doing October stuff. No, dude, it's freaking March, yo. But it's October in the Southern Hemisphere, right? It's like October. Yeah, right. You're right. I mean, you know, when you think about it, that it's like October. It's just not.
00:00:44
Speaker
You know what I mean? Does that make sense? Yeah. Cause you know, three, the March, March of the third month, October is the 10th month. Which if you subtract one, think about this, you get nine, which is three square. So it's like October. I see where you're coming from. Yeah. But before that we have some plugs and before we deliver our plugs, our website is still down. I don't know what's happening with the domain. We have to fight some pirates, but we believe that the website should be up again soon.
00:01:11
Speaker
Yeah, and if not, what we're going to do is we're going to just register a new domain and we're going to put a little note at the beginning of every episode that we have telling everyone about the new
Patreon and Merchandise Highlights
00:01:21
Speaker
domain. It'll become or some, but you know, sometimes if you're like me and you may or may not have ADHD, dealing with internet pirates is a hassle.
00:01:30
Speaker
Yes, it is. It is. Yeah. You know, when we're talking earlier, just about how October is kind of like March in a way, when you think about it, I think like, you know, if you get into numerology, maybe someday we should do a numerology episode. What do you think? I was thinking like, yeah, early April. Very early April. Let's do a numerology episode in very early April. I think that'd be a good time to do it. That'd be great. I think it'd be a very entertaining episode. Anyways, moving on.
00:01:57
Speaker
We also have a patreon patreon.com slash breaking math and I see that people are starting to help us out on that quite a bit It's helping us out a lot. Um, if you can know it's yeah, it's hugely guys. It's hugely appreciated. It really is I mean
00:02:14
Speaker
It's wonderful. We love doing journalism and our dream is to do, you know, high quality journalism, not only for math, but other subjects as well. We'd love to do that. And it's made more possible with your Patreon donations. So thank you sincerely. It's very much appreciated. It is possible to be advertisement free if we have sufficient Patreon donations. That would allow us to do what we want to do.
00:02:40
Speaker
Yeah, we want to make this an ad-free experience for you. I know when we came out with a couple of reruns, I think I accidentally baked up, put an extra ad, including the baked in ones. So I'm sorry for all you who had to experience that. But anyway, we have a tier where we deliver outlines to you. We do the episode without any ads on the Patreon. If you want it there, that's only $5 a month.
00:03:05
Speaker
to get that and We also have a Facebook poster store at facebook.com slash breaking math podcast Just click on store and we brought the price down to fifteen dollars and fifteen cents for a tensor poster huge news We now have discounted our our I should say Sofia's I didn't do a whole lot. No that is Sofia's project full disclosure I didn't have a whole lot to do with that aside from encouraging Sophia to finish it and to print it and
00:03:49
Speaker
If you're in a vector analysis class, the poster will have some fun stuff for you. If you're a physicist, you'll like it. If you like having cool things up in your office, you'll like it. I think it's 24 by 36 inches. So it's a big poster that you're getting. And we send them in poster tubes so nothing's wrinkled. And yeah. Correct. And we have now changed the price. It used to be pi to the E dollars. It is now only E to the E dollars.
00:03:56
Speaker
I did encourage you
00:04:18
Speaker
So if something to the 2 is squared, what is e to the e? It's like between squared and cubed. It's cubed. Take e and do something to it that is not really squaring, but not really cubing. It's like a fractional dimension, which you can hear more about on the Chaos Theory episode.
00:04:36
Speaker
Well, actually, I mean, just real quick, if you take something to the 2.5th power, you're squaring it and then multiplying it by its square root. And if it's 2.75, then you're multiplying that by its fourth root. And the infinity root of any number is one, so it's get closer and closer to one.
00:04:54
Speaker
But basically, yeah, that's how you generalize that. And also you can find updates on the show on Facebook. And that again is facebook.com slash breaking math podcast. And we're on Twitter at breaking math pod. And we're trying to get a little bit more active on Twitter. But send us your questions and thoughts there. We'd love hearing from you. Yes, absolutely. We certainly would.
00:05:16
Speaker
So without further ado, this is a fun episode. Again, this is a problem episode and it involves filling a human skull with liquid. So why do we want to talk about this? Do we want to talk about what inspired us to design this problem?
00:05:33
Speaker
It's a very classic calculus problem but I've always found it very poorly explained because the entire problem can be explained using thought analogy and we could do it on the podcast but usually people have to take derivatives on paper and apply it and it's actually very confusing
00:05:51
Speaker
You substitute variables and you do all this stuff for a calculus concept. And yes, you do need to know how to do that. If you're doing it on paper, you will eventually need to know how to do that.
Main Calculus Problem Introduction
00:06:00
Speaker
But if you had to learn how to write English before knowing how to speak it or something like that on paper, I'm not sure if that's a great analogy, but you know what I'm getting at.
00:06:11
Speaker
Yeah, yeah, exactly. And really, we wanted to do it because, I mean, calculus is amazing. Rates of change are amazing. Writing the outline for this episode, we just kept laughing at all of the crazy thought experiments that you can do with calculus. So I think that we were trying to, we like to make problems relatable and exciting. And I thought, well, let's do, you know, like a problem. It's exciting, like Fast and the Furious or something. But we thought, you know, what's relatable to most listeners?
00:06:40
Speaker
So, of course, the human skull. I mean, who would not want to have a human skull of their enemies? You know what I mean? Like, I certainly would. So then, you know, in designing this episode, I thought, now, wait a minute, is it even legal to have a human skull? So Sophia had to Google search. Is it legal to have a human skull? Yeah, I had to Google search. Yes, I made you Google search it. And what did you find out, Sophia?
00:07:04
Speaker
What I learned from is that in the United States at least, most states do not have restrictions on owning human body parts. The exceptions are Louisiana, where it's illegal. There's Georgia and... But everywhere else is legal to have a human skull? Georgia and Tennessee, you can buy, own, sell, and trade human remains as long as you're not going across the state border. Correct. So just be careful with that. We want to make sure we stay in the confines of the law.
00:07:33
Speaker
And speaking of the law, you also cannot in the United States own any, um, you can't grave rob anymore. You never could, but now it's really frowned upon. You can't murder, um, uh, native people. Um, their graves have been desecrated for a long time. So there are restrictions on that. Very good restrictions for that. We support that.
00:07:55
Speaker
um nature found human bones so you can't just go out into the woods and be like mine and uh viable tissue so if you cut off your foot make sure you pickle it before selling it or else it's illegal that's right you have to pickle your body parts before selling them we don't want anybody to get in trouble here on the breaking math podcast
00:08:14
Speaker
So really, maybe this episode had to deal with filling up a container of an obscure shape and maybe it just so happened that Sophia said, what about a human skull? And that was the whole reason for all of this.
00:08:29
Speaker
What does the human skull for a good reason? It's a very irregular shape. Yes. And we need a very irregular shape in what we're talking about. Exactly. Now we're assuming the human skull, all of the holes in it, except for maybe a hole in the top. First of all, we've removed a portion in the top so liquid can be poured in. And we've filled in the other holes with something like spackle.
00:08:50
Speaker
By the way, if you have a human skull and you need to spackle it, can I take a moment to recommend any of the spackle that they have at any of the hardware stores like House Depot or Bros or any of those places, you know? Bros.
00:09:06
Speaker
Yes, any spackle will do on human skull. Human skulls are remarkably porous. Yeah, just go to your local hardware store and say, hey, I need some spackle for a human skull. What do you recommend? The guy from Breaking Math sent me. So you've got a human skull. It has a hole in the top drilled, you know, or, you know, like a decent size hole, maybe a quarter size, 50 cent size. What do you think?
00:09:28
Speaker
Let's let's say big enough big enough that you can kind of look through and make measurements Yeah, like an egg will fit in that hole on top and all the other holes have been filled in very important to If you've ever seen like a sandcastle being filled in or like poured water on sand You'll notice that sometimes it'll fill up and then like there'll be like a little Dune that hasn't been filled up but once the water level reaches it all the water kind of like runs into it and
00:09:56
Speaker
If we got a bunch of pieces of construction paper and layered them within the human skull, we could just put a bunch of stacks of paper where the paper is just one sheet of paper each and fill up the cavity. Yeah, yeah. So a human skull or rather an object that looks like a human skull, it could be a human skull actually, or an object that looks like a human skull where the only hole is a hole on the top of it that doesn't ordinarily exist, I guess.
00:10:24
Speaker
Jeez, look at the, you all should be so thankful for the efforts that we go to make entertaining math problems. Okay. So we're filling up a human skull with a liquid and there's no holes except the hole in the top because of some wonderful spackle you all got from a hardware store.
Exploration of Calculus Concepts
00:10:41
Speaker
No. Okay. Now that we've established the shape of the human skull and everything like that, the question is how quickly is a water level rising at what, let's say that like we filled up, we're filling up the skull with, uh, with water from the sink and the water level will be in water or do you want some other, other liquid of your choice?
00:11:00
Speaker
Yeah, let's say we pour the black goo from the Great Abyss into the skull and we're pouring it at a constant rate. And let's say we have a little sailboat on top of it. How quickly is a sailboat rising at a given point in time? How do we measure that?
00:11:16
Speaker
Blot goo from the great abyss. Dude, that stuff sticks to your carpet like none other. This would be a great time for an advertisement on gooby gone, you know what I mean? If you have goo from the great abyss that's tracked all over your carpet, awful. No, but we do have advertisers like
00:11:36
Speaker
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00:11:55
Speaker
To support your education in math and physics, go to www.brilliant.org slash breakingmath and sign up for free. The first 200 breaking math listeners can get 20% off the annual subscription, which we have been using. And now, back to the episode.
00:12:12
Speaker
We have a human skull. The skull is being filled up with goo from the black abyss at a constant rate. Let's just say for the sake of this problem, a hundred milliliters per second. Let's say, and let's just choose the halfway point. When the skull is halfway vertically full with this liquid, how quickly is the water level rising? Okay.
00:12:34
Speaker
Now let me ask you real quickly. I'm curious when you have a very obscure shape like a human skull or you know some obscure vase or any other container. You have some information. We gave you information that you need. How the heck would you find out?
00:12:51
Speaker
what the surface area is at the halfway point. How would you know that information? Well, let's say we had a can. A can is about a circle all the way through. So all we have to do is get some calipers and measure the diameter of the can. Similarly with a vase, it is, it is radially symmetric. So all we'd have to do is measure the diameter as well. And we know the cross sectional area at every given point.
00:13:14
Speaker
However, for an irregular shape, we would have to be familiar with the entire shape inside and out. Yeah. So how the heck would you be familiar? So how would you get that information in this day and age? Would you have to do some sort of Archimedes thing where you like, you know, take a bath and put the skull under water and figure out the entire volume, then divide that by half? I mean, I
00:13:31
Speaker
I mean, it's interesting to actually bring up that area is harder to measure than volume in a 3D world. There's a machine that I've seen that does this, that you push a piece of leather through it, and it has these hooks. Not these hooks, but they're like these little gears that go through, but it's still discrete.
00:13:50
Speaker
And there's various ways that you can measure the surface area. For example, you can pour a very, very small amount of a very thin oil on top until it's, let's say, a micrometer thick, exactly a micrometer thick, and then determine how much oil you needed to fill that up and then get the area from that. But basically, I think what Gabriel's bringing up specifically too is that
00:14:15
Speaker
We're dealing now with calculus, which means that we're dealing with instantaneous rates of change and things like that, which we're going to talk about in the next section. Yeah, yeah. Now, let's just say in respect to the skull, just because I'm curious, in order to find out the halfway point, you'd have to be like a dolphin and use echolocation or something like that, right?
00:14:36
Speaker
Yeah, you know your dolphin you won this battle the skull in a battle of wits with your fellow dolphins You're ready to take over the world. You're filling it up with the goof in the black abyss You're using that in your generator and you want to know how quickly it's rising because you have reasons So a dolphin would be able to using echolocation would be able to figure out the exact halfway point Yeah, so we're assuming basically the point all this is that we're assuming perfect knowledge about the skull and
00:15:00
Speaker
OK, very good. So we know the cross-sectional area at every height. I'm not meaning to be such a pain, but it just makes me curious. How would one figure out this information? But the audience, all you need to know is that you have that information, whether you're a dolphin or whether you're some sort of interdimensional being, you have that information. So without further ado, we are moving on. Use an MRI machine. Yes, use an MRI machine. All right. So this next section of this podcast, we're talking about this problem involving
00:15:29
Speaker
filling a skull with a liquid, which we've already identified. Let's talk about something called instantaneous rates of change. Now this is an important concept for anybody, even high schoolers, who have a basic knowledge of calculus.
00:15:46
Speaker
Now, they do these math problems and if I may be so bold as to say, our boring is all get out. You study instantaneous rates of change, but how often do you really understand what it is that you're doing? So let's take a moment and talk about instantaneous rates of change.
00:16:03
Speaker
All right. Yeah. So, um, who's, who's the character from fashion? The furious. Oh gosh. I I've never seen, I've never seen any of those. Let's just say that the, I I've never seen it, but let's say that it's called max furious max. You're good. You'll all see, you know, max furious. So max furious is trying to beat up the drug dealers or whatever he does. And he's in a car and it speeds up from a dead stop.
00:16:28
Speaker
Are we gonna lose listeners because we're making up stuff that's not in Fashion the Furious? This is a math podcast. This is not a Fashion the Furious podcast. And for all of you wonderful fans of that movie, I'm sure it's phenomenal. I'm sure it's beautiful. And I know that we're about to slaughter it. Oh my goodness, we're about to slaughter it. But it is all in the name of explaining a math concept, specifically instantaneous rates of change.
00:16:53
Speaker
So I'm bracing myself for the brunt of hateful emails and hateful reviews that we're about to get. So anyway, Max Furious starts with zero velocity and that increases over time. So you can imagine this car, it has a speedometer like any other car. And Max Furious, the hero of Fast and the Furious, looks at the speedometer at like, let's say like 3.35 PM. And let's say it says 40 miles per hour. That is his instantaneous rate of change.
00:17:22
Speaker
And this is crazy. Instantaneous rate of change. So one of the things that is really mind-blowing about this is we've spoken before about calculus and you have your position, which is, how do we explain position? It's like where we start with, right?
00:17:42
Speaker
Yeah. Position is like where, how, how much distance there is between you and your starting position. So in calculus, there's a lot of concepts here, including position, the derivative of position. You got to use calculus for this. The derivative of position over time is velocity or your speed.
00:18:00
Speaker
And if you notice the units, position is meters or like feet or whatever, and time is like hours or seconds or whatever. Yeah, so like three feet per second. Yeah, so the derivative of something over something will have units of that first thing over the second thing. So the derivative of position over time will be in meters per second.
00:18:22
Speaker
You know, let's say you're doing the derivative of position over color. Like you just want to, like you want to see how much the position changes of stuff based on the color of the environment or something like that. Then it will be in the bizarre units of position per color. Yeah. Yeah, exactly. So basically don't know how to interpret that.
00:18:39
Speaker
We're just talking about keeping our units straight because everything we're talking about here, everything, the consistent idea of anything meaningful at all is one thing in relation to another thing. And in this case, it is your speed in relation, I'm sorry, it is the, I'm sorry, what? So with position in relation, so
00:19:01
Speaker
So you're saying that, yeah, that the derivative of a position with respect to time is equaled in units of position over the units of time, which is in meters per second or miles per hour. Yeah. And if you notice with some
00:19:17
Speaker
Oh, yeah. Well, I was just going to say that the thing that really got me thinking was you have position. The derivative of position is velocity. The derivative of velocity is the change in velocity over time is acceleration. The derivative of acceleration is jerk. Right before we go into that, the reason why the derivative of velocity is acceleration is because you can imagine like in like let's say right now you're going 40 miles per hour and in one second you're going 50 miles per hour.
00:19:46
Speaker
That means in one second, you increased 10 miles per hour. So your acceleration was 10 miles per hour per second, which works out to something better. You can multiply hours in seconds, but I don't feel like it. Yeah, no, I understand. My issue is that you don't necessarily always have the same acceleration. You can have a variable acceleration. Like let's say someone has a very unsure foot on the acceleration and the car lurches backward and forward and backward and forward.
00:20:16
Speaker
like on the interstate, that means that the acceleration is constantly varying. And actually it's interesting is that the amount of lurch that you feel is exactly proportional to acceleration, which is actually something that Einstein used in his second theory of relativity.
00:20:36
Speaker
I didn't know that I said had a sequel. Oh yeah. General, the general theory. Well, I knew that. Oh, second theory relativity is the general theory. I knew that. I knew he had two theories, but I didn't, I've never heard it in terms of like the sequel. Oh yeah. I have never heard it either. I just do things sometimes. I just say things. Okay. Okay. This is funny. This is funny.
00:20:54
Speaker
Yeah, so anyways, the reason why I brought all that up is because there's this concept that, a thought experiment that I think that we talked about this in a previous episode called Zeno's Paradox, where you talk about the difference between something at rest and something in movement. What is the exact moment that you begin to be in movement?
00:21:18
Speaker
And that's a very hard question to ask because it's a continuous thing and continuous things are infinite. You know what I mean? Oh, yeah. Yeah. You're dividing infinitesimals by infinitesimals to get like two miles per hour, like type numbers.
00:21:34
Speaker
It's fun. The foundational aspects of derivatives is you take a thing called a limit, which is like, let's say that at 305, you're somewhere and at 310, you're somewhere else and you calculate the average velocity.
00:21:51
Speaker
Well, if you did it at like 305 and then 306, you get something much closer to what the instantaneous was rate at what 305 would be. And then at 305 and 305 in one second, you get even closer. And then the derivative is like, if you kept going forever, what would you get?
00:22:07
Speaker
Correct, yeah, yeah. Now, obviously all of this relates to filling up an obscure shape, and for the purpose of this episode, a skull with a liquid and the rate of change of
Calculus in Simple and Complex Shapes
00:22:17
Speaker
that. But still, calculus, no matter how many times I am now 35 years old, in my 35 years, I never, ever cease to be amazed at calculus as I'm trying to explain it to somebody. It always blows my mind. I also have to point out that Gabriel looked at his watch when he was thinking about what age he was.
00:22:36
Speaker
Yeah, oh, why did I do that? I don't exactly know why I did that. Before we talk about the rate of change of a liquid that is filling up an obscure shape such as a skull, let's talk about a much, much, much simpler problem. Sophia, do you want to introduce a simpler problem?
00:22:54
Speaker
Sure, let's say you have literally a box that you're filling with water from a tap. The tap is dispensing 100 milliliters of water per second. And remember 100 milliliters is 100 cubic centimeters. And let's say that the box, let's say it's pretty tall, whatever height, it doesn't matter what height it is, but let's say it's 10 centimeters wide and 10 centimeters deep. So it has a cross sectional area of 100 centimeters squared.
00:23:21
Speaker
That means that, so let's, let's say, let's say that after let's say we're pouring in water, right. Const at a constant speed after one second. How many, how much water has been dispensed? Well, that's a pretty easy problem then. And it's easy because we've got a consistent shape. It's the same shape all throughout.
00:23:37
Speaker
Oh yeah, but I'm saying if the tap dispenses 100 milliliters of water per second, after one second, we have how many milliliters of water? 100 milliliters. Yeah, which is 100 centimeters cubed. And remember that the definition for the volume of something is how high it is times the cross-sectional area it is, if it's like a perfect extrusion. And by extrusion, I just mean imagine making something out of Play-Doh and pressing it through something.
00:24:05
Speaker
So 100 centimeters cubed divided by 100 centimeters squared, which again is the rate of flow divided by the cross sectional area is one centimeter. So basically, and since that happened over one second, the water is rising one centimeter per second. Yeah, this with this particular problem, you don't even need any calculus, any calculus whatsoever.
00:24:26
Speaker
And if you think about it, like, let's say this was like a tiny, like a really skinny, like square, like tube thing, and it's one centimeter by one centimeter. It would rise a hundred centimeters in one second, right? Yep. And that makes sense. It could be like versus like,
00:24:42
Speaker
Exactly. So all we have to do if we know that we're filling up a box from a tap is take the cross-sectional area of the box and take that and have it divided by the rate of flow. Yeah, exactly.
00:25:02
Speaker
there's going to be a difference between the very simple shape like this shape, you know, a cylinder or even a cube and the skull. So let's briefly talk about how we would do the same problem with a slightly more complex problem where you've got something that's not so simple as a cube.
00:25:23
Speaker
Yeah, oh yeah and before we do that I want to make a note that the box doesn't have to be it doesn't have to be a box if that had been a cylinder all we have to know is that we don't the cross-sectional area to fill up the cylindrical tube you can also imagine it like imagine we had a bunch of pancakes with I don't know Totoro on them and you have a stack of Totoro pancakes
00:25:46
Speaker
and you make a tube that is Totoro shaped. All you have to know is the area of Totoro to know how quickly the water level is increasing in this Totoro tube. I've never heard Totoro until you mentioned it just now. I just got into Hayao Miyazaki. He's this family film director from Japan who designs these visually incredible films. I know I'm very late to the bandwagon, but
00:26:12
Speaker
Oh wow, okay, very good, very good. So, talking about a slightly more complex problem, I'll go and read this one here. Suppose you have a cone that is 100 centimeters squared at the top and is one meter tall. When it is at the vertical halfway point, how fast is the water level rising? We can conceptualize this cone as a bunch of cylinders of increasing size.
00:26:39
Speaker
Yeah. I was like, kind of like a, like one of those things that scoops honey out. I was thinking, or like Devo. I don't know what either. What's Devo? Devo the band from the eighties. Okay. They had these hats that were basically like a bunch of like smaller cylinders on top of each other. I was not aware of that. I'm going to have to Google search Devo hat. I'll do it right now. Okay. Hang on. I'm going to Google search Devo hats real quick.
00:27:02
Speaker
Oh my goodness. Devo hats. They're the red things that they're ridiculous. I'm sorry. We should edit that part out. Whatever. I don't care. I call them ridiculous. No, Devo would like you calling them ridiculous. Okay. Interesting. Yeah. It looks like a Lego thing. There needs to be a Lego version of Devo. Lego Devo, I think. It's like Russian dolls almost. Dude, all listeners do yourself a favor. Google search Devo hats. Let's try to make that go viral. Let's bring those back.
00:27:31
Speaker
So yeah, you can imagine like you have, like you have this cone, right? And you got to, you can approximate it as like at the very, very bottom of the cone, you can have this tiny little cylinder, maybe the size of a pencil eraser. And then right above that you have like a disc the size of like a few stacked coins. And then the next, the size of a few more stacked, larger coins and you have, yeah. And they have this bigger and bigger thing until you have this CDs. How about can we do CDs?
00:27:56
Speaker
Yeah, it goes to CDs, then it goes to laser discs and, you know, to the whole like, yeah. Like car tires, maybe a four flatter car, like pieces. Yeah, although then we're getting way above 100 centimeters squared. Oh. But yeah, exactly. It's the same vein. You have bigger and bigger cylinders and together they look like a cone. Yes. So when the water's at the halfway point, it is filling up the cylinder that is at the halfway point.
00:28:21
Speaker
And it is doing that. And remember, Akon, the width at the halfway point is going to be half the width at the point where it's the widest. Yes. So and since we're doing area, we have to square it because you go back to episode one and listen to Mino's slave.
Analysis of Real-World Applications
00:28:42
Speaker
Talk about how you have to take a half, like taking a quarter or something kind of feels like taking a half sometimes. But yeah, it's basically 25 centimeters squared. So if we know that we're putting in water at a rate of 100 centimeters cubed per second, and this has a cross sectional area of 25 centimeters squared, how quickly is the water level rising?
00:29:05
Speaker
Okay. So from that information, we know that the water is rising at an instant rate of four centimeters per second. Yeah. And if you think about it, that's, it's, it's a lot faster than it was at a hundred centimeters squared. And that makes sense, right? Because when you fill up a cone, it would go really quickly at the beginning and then get slower and slower and slower and slower. So it would be, so the rate of change. So the derivative here, the, the acceleration would be a constant slowing down.
00:29:31
Speaker
Yeah. And what's interesting too is that, um, the bottom half of the cone would take something like one eighth of the time to fill up, uh, the bottom half of the cone, because, um, we would have to cube the one half. So we're dealing with different dimensional stuff here. I just wanted to point that out. Thank you. I'm glad you pointed that out.
00:29:53
Speaker
So we're about to get weird here. We're about to get a little weird. We're gonna go from a cone to the strange shape of a human skull. Now a human skull is a varying shape. You have areas of probably, you know, as you go from the bottom up, generally it gets bigger, but not necessarily always.
00:30:13
Speaker
Yeah, especially with the human skull, there's a lot of weird cavities within the skull, like a lot of sinuses and stuff like that. And we're assuming right now that we're only carrying out the part of the skull that, like let's say we had a bunch of, like we said earlier, a bunch of pieces of cardboard or something like that, we stacked them up on top of each other to fill up the cavity we're talking about.
00:30:33
Speaker
No, let's say that we wanted to know how quickly the black goo from the Great Abyss is filling up the skull vertically when it's at a certain point. What's our goal? Let's just say we filled up a skull with the black goo from the abyss. What are we going to do with it then?
00:30:51
Speaker
Gabriel, we have no need for money from the human realm. We just need the skull with the black goo. Okay, I've got a friend who sells artistic skulls. Her name is Leanna, actually on Facebook. She has an artistic channel called Something In Between. And you guys should check out Something In Between and see all of her skull art and all the black goo. Her and her husband, Matt Stolfitz, pretty cool stuff.
00:31:19
Speaker
Yeah, but let's say, yeah. So we looked at the Visible Human Project. We found a video on YouTube. It's this thing where they sliced a frozen human cadaver at like, I think, 1.3 millimeter intervals. And we found out that the cross-sectional area
00:31:37
Speaker
of the human skull at about the point where the nose is, is about 300 centimeters squared. So we're filling it up with black goo from the abyss, from some kind of abyss tap, and it's flowing at 100 centimeters cubed per second. It is 100 centimeters cubed per second divided by 300 centimeters squared, which means that the water level would be rising at about 3.3 millimeters per second.
00:32:01
Speaker
Correct, so in order to solve for an obscure shape, you just have to have, when you look for an instantaneous rate of change at that instant, what's the rate of change? Yeah, and if you imagine that we had some kind of tap that was delivering a whole liter per second, then at the nose point where it's 300 centimeters in cross-section, it'd be filling up at about 3.3 centimeters per second.
00:32:28
Speaker
It's not possible to solve this problem unless you do have the fact that you're at the halfway point. Yeah, but we have the advantage of being mathematicians where we know everything within our world. Correct. Yes, yes, exactly. So assuming you know it is in fact the halfway point, that's the information that you need.
00:32:45
Speaker
Or whatever point you care about. It doesn't have to be halfway pointed. Yeah. If you wanted to see like, if it was like when it reaches the eyeballs, how quickly it'd be increasing. I don't know. I actually, that's one thing about this problem that I never understood from calculus is why anybody would care about how quickly the water level is rising.
00:33:00
Speaker
Yes, I can't think of a reason why that would matter. And if you're if you're about to die and you want to calculate how much oxygen you have left for some time, it would be volume. OK, the water level rising would be like if you had like a boat and somebody please email us and tell us why the heck would you care how fast the water level is right. It was rising. We're not talking about volume here. And we're actually interested if you are an engineer and you do need to know how quickly the water level is rising within a tank for some reason. Please let us know. We'd be very interested in hearing.
00:33:29
Speaker
Yeah. And we're not just talking about, so, you know, you know, when it fills up, because again, that's a volume problem. We're specifically talking about the rate of water level rising. I don't know. I don't, I can't think of a reason for more information on what we're talking about. Listen to the rest of the episode, the previous bunch of the episode, some further problems. Now here is an interesting spin on the above problem.
00:33:55
Speaker
What if the black goo from the abyss, what if it is changing the flow rate over time in a predictable way? Or somewhere that we at least know. Hence predictable way. Oh yeah, but I mean like in the mathematical sense where it's not actually predictable, we just happen to know what it'll do. I guess it's predictable. Okay, and we want to know how quickly the water is rising at a certain level. We need to know the volume of the skull below that point. Like the eyeballs or the nose or whatever.
00:34:23
Speaker
We need to know the volume of the skull below that point, the point at which that much black goo was delivered by the black goo delivery system. I mean, how does that work? I don't know about the plumbing with the abyss, but whatever. That's not my... Well, this would be the integral of the black goo rate function over time. Okay. The integral is like adding up like little sizes. Exactly. Good. Good.
00:34:46
Speaker
So to get the rate of the GU addition at that point in time. And we need to know the cross section area at the level that we are concerned
Episode Conclusion and Reflections
00:34:55
Speaker
with.
00:34:55
Speaker
Yeah. So basically what we've shown here is that we get one rate of change from another rate of change and we've done it with a simple word problem. Yes. So all of you who, who are wondering, you know, what do you need math for? Well, your answer can be, you now know how to find the instantaneous rate of change of black goo when you're filling up a skull. And you know, if everything goes to plan, that'll matter quite a bit. Yes, it will. I'm Gabriel.
00:35:25
Speaker
And I am Sophia who usually goes first, but I stole the first spot there. So are we done? Like, should we add some evil laughter or something? Like, um, wait, where's that evil laughter?