Related Rates
October 14, 2026

[00:00:05] Okay, let's pretend this balloon is a perfect sphere.
[00:00:10] Okay, that was exciting. So, over those, what was that, about eight seconds? Over those
[00:00:21] eight seconds while I was blowing up the balloon. What was changing? Hold on, hold on, let's
[00:00:31] go one at a time. Somebody from that row. Volume change. What else? Sorry, let's go second
[00:00:38] row. Surface area is changing. What else? Sorry, this row. Anybody from this row? The
[00:00:57] diameter change. Somebody from this row. The radius change. Somebody from this row.
[00:01:11] The size. Yes. What size? The size of the... Sure, let's go with circumference. That's another one.
[00:01:26] What else changed? Somebody from this row. Oh, is that it? Did the color change at all?
[00:01:47] The color changed a little bit. How about the thickness of the balloon, the latex, whatever it is.
[00:02:09] The air in my lungs changed. Volume of lungs. Anything else? I'm sure we can come up with more. But lots of things change over time.
[00:02:22] As I release air, all these same things change. So, lots of things change. They're all related in some way. So, we'll get to that in a second.
[00:02:48] So, if it's a sphere, what's the formula for volume of a sphere? R cubed. Somebody discovered it. We're actually going to prove this formula later on.
[00:03:14] So, this equation is how volume is related to radius. They're related by exactly this formula. At any given time, if we knew the radius, we could find the volume.
[00:03:36] At any given time, we knew the volume, we could find the radius. So, that's how these variables are related. These are also functions. So, V is a function and R is a function. It's not just a variable. What is the volume a function of over those eight seconds?
[00:04:00] Is it a function of time? We could write it like that. Because the volume changed over the time period, it's a function of time. What is radius a function of? Time as well. So, even though we just see variables, they are functions.
[00:04:24] And they're functions of time as time passed. The volume changed, the radius changed, and so they're functions.
[00:04:37] Now, if we want to know how the rates have changed, how fast the volume is changing, or how fast the radius is changing, we simply differentiate. Using what kind of differentiation?
[00:04:53] Implicit. Implicit, because we've got functions on both sides. So, we differentiate V. What is that? With respect to time, that's simply dV/dt. We differentiate this, we get 4 3rd pi times 3 R squared. And because of the chain rule, times...
[00:05:23] The derivative of R, which is dr/dt.
[00:05:31] dV/dt is the rate of change of volume with respect to time. It's how fast the volume is changing.
[00:05:45] And this is how fast the radius is changing. And how are the rates related? They're related by exactly this equation. It's that simple. So, this is how the functions are related, or the variables. This is how the rates of those functions are related.
[00:06:08] Okay, so related rates is simply an equation that involves d derivatives.
[00:06:17] Okay. As I was blowing it up, let's say I was blowing it up at 1 gallon per... Let's not 2 gallons. Let's go inches cubed per...
[00:06:41] 20 seconds. Let's assume it's a constant rate. Let's assume it didn't change for a minute. That would be our function dV/dt. dV/dt is a function. It's a function of time as well.
[00:07:01] And the units for any rate of change is always the units of this variable or function divided by the units of this variable or function. Okay. That's why it's inches cubed per seconds or something.
[00:07:22] Is this a positive rate or a negative rate? I mean, obviously I've wrote it as positive. But would it be positive or negative?
[00:07:35] It is positive. Why is it positive? Yeah, so whenever this function... If V is increasing, it's a positive rate. If volume... Like when I let out the air at the end, volume was decreasing, it would not be this.
[00:08:00] It would be a negative rate. Okay. If I let out the volume at the exact same rate I blew it in, then it would be exactly dV/dt. It would simply be negative. 1 inch. Oh, that's not much. 1 inch cubed, but that's okay. 20 seconds.
[00:08:27] It works better with gallons. But that would be a very slow rate, right? One cubic inch per twenty seconds. Anyways, whenever things are changing over time, if the function V or r is increasing, the derivative is positive. The rate is positive.
[00:08:49] If the function is decreasing, like the rate is decreasing, it would be a dr, it would be a negative value. Okay, so sometimes they're just going to give you numbers for a rate. And we have to decide... They'll give you a positive number. And they'll say whether it's increasing or decreasing or you have to figure it out.
[00:09:11] And that's going to be very important, whether it's positive or negative. Surface area. What's the formula for surface area of a cube? I actually don't know what it is. Sorry, did I say a cube? I meant a sphere.
[00:09:37] Is it 4 pi R square? Yeah. Okay, I'm trusting you. Again, this is how surface area is related to radius at any point in time. They're both functions of time. We could write it as surface area of t.
[00:10:04] R of t. Okay, because they are functions of time, even though we just write them as a variable. And we differentiate this.
[00:10:14] This dS/dt is four—so that is eight pi r to the first, times the derivative of r, dr/dt. This is how the rate of change, how fast the surface area is—
[00:10:44] How fast the surface area is changing is related to how fast the radius is changing. If we were given one of these, like how fast the radius is changing, the rate of change of surface area changes at every point in time based on this and the given radius at that moment.
[00:11:14] What other equations do we have? We probably could not come up with a solution here. Again, let's say, what would be the units for surface area?
[00:11:37] Yeah, whatever the units, let's call it inches squared. And what would be the units for radius? Just be inches. Okay. The units for this are going to be what? Inches squared per whatever the units of time is. Let's call it a second.
[00:12:01] What would be the units of this? Inches, just inches is radius in seconds. And what are the units of the R? Inches. Okay. And you can see an inch times an inch divided by a second is inches squared per second.
[00:12:23] It always works out that way. And we're typically not going to put in units until the end. It just makes it way too complicated. I'll show you how we do that later on.
[00:12:43] Okay.
[00:12:46] Can we come up with a formula that relates surface area and volume?
[00:12:54] We could. We would need to take this equation, volume is four thirds pi r cubed, and replace this r with that r.
[00:13:11] Okay. So what would be three-fourths V? Let's go three over pi equals R cubed. And then take the cube root of both sides.
[00:13:32] So we get r is the cube root of 3V over 4 pi. Now we replace this with this r right here. So volume equals four thirds pi r cubed.
[00:14:02] Wait, sorry. Sorry, I meant to replace it here. Let's delete that. So surface area equals 4 pi R squared, which is the cube root of 3V over 4 pi.
[00:14:26] Okay. So that's, we can often come up, no, often come up with relationships that we rarely ever deal with. But this is exactly how surface area is related to volume at any point in time.
[00:14:41] If we differentiate this, we're not going to, what, this is kind of messy. Actually, it's not that bad. But anyways. That would give us the rate of change of the surface area. How is related to the rate of change of the volume?
[00:15:00] Okay.
[00:15:02] Times the height. Okay. Again, this is how the volume is related to the both the radius and the height. Or this is how the height is related to the volume by this equation. At any point in time, if we knew two of them, we could find the third one.
[00:15:23] Again, these are functions. They're all functions of time. So if we differentiate, we get dV/dt is, what rule do we got to use here? Product rule.
[00:15:39] Two pi r times dr/dt times h, plus pi r squared times the derivative of height, dh/dt.
[00:15:58] So again, this is how all those rates are related. We're filling this up with water. How fast the volume is changing this. This is how fast the radius is changing. This is how fast the height is changing.
[00:16:14] And again, over time, they all depend on all of them, including the radius at that point in time, including the height at that point in time.
[00:16:30] Um, what's the formula for volume of a cone?
[00:16:39] It is kind of weird, but it is exactly one third the volume of a cylinder: pi r squared h. We are going to do a lot of these, because we are filling this up with water.
[00:16:58] Now the radius at any point in time, whatever the height of the water is, the radius is changing. The height is changing and the volume is changing. So that makes it fine.
[00:17:17] The derivative looks exactly like this, except we have a one third out here. Okay.
[00:17:29] Okay. The other thing with cones, right, let's say this is the R. Let's call this big R for a second. Let's call little r the radius at any given time.
[00:17:49] Let's call this big H, the total height of the cone. The other equation we could always come up with is how H and R are related. How are H and R any, sorry, let me do a little h here.
[00:18:07] How are they related? Sorry, if capital R.
[00:18:18] So we have two triangles here, right?
[00:18:23] We got the triangle of the volume and we got the triangle of the big of the cone. Okay. R and H are fixed. R equals some number. H equals some number.
[00:18:44] And they're proportional triangles. Very important. So R over H is equal to what? Little r over h. Okay. And let's say it was, the height was 10 feet.
[00:19:05] Let’s pick R, and H is six feet. We know ten over six is equal to little r over little h, and we multiply by h.
[00:19:25] What is it? Five thirds.
[00:19:33] Equals R.
[00:19:35] So almost all cone problems. The radius and the height of the cone are some fixed number.
[00:19:49] We come up with that, and then we differentiate this, which is simply five thirds times the derivative of the height of the water.
[00:20:00] And this is the R. Okay.
[00:20:08] And then once we get our first real problem, we'll show you what we do with that.
[00:20:16] Anything that was triangles, we come up with lots of equations.
[00:20:24] The triangle has to be changing in some way. Let's say this is some person and he's walking away. And this is another person.
[00:20:40] The triangle has to be changing. The green will be my new triangle.
[00:20:49] Let me call this theta for a second. So what equations can we come up with with the triangle?
[00:21:01] I don't see A's or B's. I like that though. There we go.
[00:21:08] And why is that true? Good old Pythagorean, our favorite geometer.
[00:21:21] Okay. This is how those four functions are related. Or sorry, three functions, three variables.
[00:21:29] If we differentiate, this one is easy: two x times dx/dt plus two y times dy/dt equals two z times dz/dt.
[00:21:51] And this is how their rates of change are related.
[00:21:59] Okay. If we wanted to figure out how fast the distance between the two guys are, that would be this. We divide everything by 2z. Oops.
[00:22:15] What is that? Two x dx/dt plus two y dy/dt, over two z.
[00:22:56] Okay.
[00:22:59] This is exactly the units of this divided by the units of that. Okay. So if we were to plug everything in, we don't plug in any units. We just know at the end, we look at the units of z divided by the units of t.
[00:23:11] And that's the units of this number. Okay.
[00:23:16] What other equation can we come up with this triangle?
[00:23:31] How about area?
[00:23:35] One half times base times height. Let me just call it base times height for a second. Or let's call this the base. This is the height. You could use x and y or change the variables.
[00:23:50] Okay. So this guy's walking. This area is changing. It's not that, it's kind of a weird thing, but this is how the area is related to the base, related to the height at any point in time.
[00:24:06] We differentiate dA/dt. And what rule we got to use here? Product rule again.
[00:24:15] So that is one half db/dt times the height, plus one half the base times the derivative of h, dh/dt.
[00:24:29] This is how the rate, how fast the area is changing. It depends on how fast B is changing. Depends on the height of that given time. Depends on the base at that point in time. It depends on the rate of change of the height at any given time.
[00:24:44] We got an angle. In fact, we got three angles, right? We can come up with lots of equations involving this.
[00:24:54] We can come up with sine of theta. What would that be? X over Z.
[00:25:07] What is cosine of theta? Y over Z.
[00:25:17] What is tangent of theta? X over Y.
[00:25:22] We could obviously do secant, cosecant, and use the other derivatives. Usually we use only one of these three.
[00:25:29] This is how the angle is related to the other two sides of the triangle.
[00:25:40] If we differentiate, let's start with this one. We get, what's the derivative of theta, or sine of theta?
[00:25:50] Cosine of theta times the derivative of theta, dθ/dt. That is the left side. What rule do we use on the right side? Quotient rule.
[00:26:04] dx/dt times z, plus x times dz/dt, over z squared.
[00:26:23] Let's say we wanted to know how fast the angle is changing. We just divide both sides by cosine. That would be that equation.
[00:26:39] Did I screw up? Yes, this is definitely a minus.
[00:26:48] Let's say we do all the variables here. We plug them in. What are the units of this going to be?
[00:27:05] This is very important. In calculus, this is always in radians. If they ever give you a value in degrees, we need to convert it to radians.
[00:27:19] Except the only thing you wouldn't need to do is you can just find it there. It's guaranteed to be something radians per whatever time it is.
[00:27:32] And really radians don't have units. Why do radians don't have units?
[00:27:54] So that's some theta. Anybody remember how we defined the angle measurement in radians?
[00:28:14] Theta is simply the ratio of the arc length over the radius. And both of these are distances.
[00:28:26] Let's say it was inches. The inches cancel out and there are no units. So radians don't really have units. Sometimes we don't even write radians, we just put nothing per second.
[00:28:42] And that's okay. But we know it's radians. It's going to be very important. We always need to know the things in radians.
[00:29:00] What is the first graph going to look like? It is going to be some linear function. The rate of change of volume is dV/dt.
[00:29:14] Okay. What about the one on the right?
[00:29:24] So what happens once it gets here?
[00:29:30] Yes, the height grows a rate much faster. So it's going to be something like this.
[00:29:39] [Brief speech unclear.]
[00:30:01] It would be the height.
[00:30:07] Something like that. Third one.
[00:30:17] So slow.
[00:30:22] Faster, slower, something like that.
[00:30:26] Okay. What about this one?
[00:30:31] So faster and slower, something like that.
[00:30:35] Right. Three inches a year. Three inches a year.
[00:30:47] That's a lot of it. It depends on their point in time. Any time you have a rate, it's a d something dt. Let's change it over time.
[00:31:02] So this would be d what over dt?
[00:31:09] Inches is what kind of thing?
[00:31:13] It's a distance, but when you're growing, we call that your height. So it would be d h dt.
[00:31:24] And is it positive three or negative three inches per year?
[00:31:29] Because we're growing, it's positive three inches per year.
[00:31:40] My mutual fund is shrinking. What does that tell us since it's shrinking? Negative.
[00:31:48] Four cents per day.
[00:31:51] And this would be d what? Dt.
[00:31:57] Okay. It helps if you look at the units. What is the units? What is cents a measurement of? Money. So you could call it d money, dt.
[00:32:10] D fund, dt. D value, dt, whatever the value of the fund is. So this would be, it's called dm dt.
[00:32:23] The radius of a circle is increasing.
[00:32:28] So that would be what? D radius dt. Is it positive or negative four feet per hour? Positive.
[00:32:46] The volume of the cone is decreasing. So that would be d v dt.
[00:32:57] And is it positive or negative?
[00:33:00] When you tell if it's decreasing, negative two inches cubed per second. Again, we could look at the units.
[00:33:10] Inches cubed is a measurement of volume.
[00:33:13] So in seconds is a measurement of time.
[00:33:19] Okay, next page.
[00:33:20] We got a rectangle. Start by just drawing a rectangle.
[00:33:26] I don't know if there's much room here, but.
[00:33:31] Okay.
[00:33:33] Got a width. We got a, what else we got? So just call the width and the length. Whatever you want to call it. Let's call this the length.
[00:33:47] This is the width.
[00:33:50] Okay.
[00:33:52] And then we come up with some, some, how those two variables are related. And what are we talking about the, what's the question?
[00:34:06] Perimeter. Okay, so we come up with an equation involving a perimeter. What's the equation?
[00:34:16] Perimeter equals. 2L. 2L plus 2W.
[00:34:24] And then I automatically just differentiate that.
[00:34:29] So derivative of perimeter with respect to time. Is 2 times the derivative of L with respect to time. Plus 2 times.
[00:34:43] Okay. That's how their three rates are related. Okay, then I, I go back through the problem and label stuff.
[00:34:55] So it's the length is increasing at a rate. So what is that going to be?
[00:35:05] L. DT. Anytime you have a, they say the word rate. It's a D something DT. And since it's increasing, it is positive 2 inches per second.
[00:35:22] And the width is increasing.
[00:35:26] At a rate, that's going to be a D what?
[00:35:31] D. W.
[00:35:35] DT.
[00:35:42] Equals 3 inches per second.
[00:35:50] Okay, then how fast? Same thing the rest of the facts. They're asking for a rate. Okay.
[00:35:57] Perimeter. So we want to know. D. P. DT.
[00:36:03] Okay.
[00:36:06] And then they give us exact moments of time. We never plug these in. Until after we differentiate.
[00:36:15] Okay. Never write this formula with any numbers in there. Unless we're told one of those sides is not changing.
[00:36:23] Or you know it's not changing. Okay. So.
[00:36:30] What would this be?
[00:36:34] This is L.
[00:36:37] Of the length. At some time, unknown time.
[00:36:45] Equals 10 inches. And the width at some point in time, the same point in time.
[00:36:53] Is 6 inches.
[00:36:55] So we want to know D. B. D. So we solve for that. So we solve. So it's simply D. D. P. D. T. Is 2 times. What is D. L. DT.
[00:37:12] We wrote up here.
[00:37:14] Plus 2. Times. D. W. DT. Which is right here.
[00:37:24] And again, don't bother putting in units. It gets super messy if you do. We got 4 plus 6 is 10. Okay. Now we put in units.
[00:37:34] And the way we figure out is you put in the units of P divided by the units of time. The units of P was.
[00:37:41] Inches. To distance and units of time was seconds.
[00:37:47] So at that one moment in time.
[00:37:51] When the length was exactly this and the width was exactly this. The rate of change of the perimeter is exactly this. Okay.
[00:38:00] Actually, did we even plug in those numbers at all? No. Do we need you?
[00:38:07] No. Okay. So in fact, in every moment in time.
[00:38:13] Not just a specific moment. It happens to not change. And the reason is that the derivative doesn't involve L's or W's.
[00:38:22] If this were like W squared, we would. Definitely have a W in it. And then it would depend on that one time. Okay.
[00:38:31] So in fact, this is a constant rate of change.
[00:38:35] Because these two were constants.
[00:38:38] Interesting.
[00:38:41] Now the rectangle's length is increasing at 3 inches.
[00:38:49] So the rectangle's length. What would that be?
[00:38:55] D L.
[00:38:58] D T. Is increasing. So it's 3 inches per second.
[00:39:06] The width is decreasing. So D.
[00:39:11] W. D T. H. Is very important. Negative 2 inches per second.
[00:39:23] Area. We need new equations.
[00:39:26] What is area of a rectangle? Length times width. Okay. Now we got to differentiate.
[00:39:35] D area. D T.
[00:39:40] Is D L. D T.
[00:39:45] Plus.
[00:39:48] L.
[00:39:51] Times. D W. D T.
[00:40:01] So D.
[00:40:08] D A. D T.
[00:40:14] Is. What was this rate? That's 3.
[00:40:18] Okay. Times W at that one moment in time, which is 6.
[00:40:26] Plus.
[00:40:28] L. Which is 10 at that one moment in time. Times D W D T. Which is negative 2.
[00:40:48] What is that negative 2.
[00:40:57] And the units are. The units of A divided by units of time. What is the units of area?
[00:41:05] Everything was in inches. So it's inches squared. So.
[00:41:12] Time was measured in seconds.
[00:41:16] Okay.
[00:41:20] What does it mean that this is negative?
[00:41:24] It is getting smaller. It means the area. The area is decreasing at a rate of two square inches per second.
[00:41:31] Okay, spherical snowball melts. At the instant—one moment in time—at which the radius is twenty centimeters.
[00:41:44] Its radius is decreasing at three centimeters per minute. At what rate is the volume of the snowball changing?
[00:41:57] So what formula do you come up with?
[00:42:02] We went over volume or the rate of change of volume. So volume equals volume of a sphere, which we talked earlier.
[00:42:15] We might as well label we got. What is the 20 centimeters?
[00:42:28] That's the radius at one moment in time. So that's our of some time. Let's call it a is 20.
[00:42:40] Its radius is decreasing. What is this?
[00:42:46] Dr.
[00:42:48] Dt is a positive or negative negative three centimeters per minute.
[00:42:55] And we want to know the rate of the volume that is dv dt.
[00:43:09] We differentiate this, which we did earlier, right?
[00:43:13] dV/dt is four thirds pi times three r squared times the derivative of r, dr/dt.
[00:43:34] So dv dt at that one moment in time. I called it a earlier. We don't know what it is. Doesn't matter.
[00:43:50] What was the radius of that time?
[00:43:56] Twenty.
[00:43:58] And dr/dt was negative three.
[00:44:03] And we do some math. What's this going to be?
[00:44:23] Nothing negative.
[00:44:30] Four hundred times four is sixteen hundred; times three is forty-eight hundred.
[00:44:48] And we put in pi. What did you say? Negative what?
[00:45:02] About three times that seems about right. Point what?
[00:45:09] So most of the time when it's a calculator problem, we need to leave three digits. Give me three digits: six four five.
[00:45:24] So this is the exact rate of change. This is an approximate rate of change.
[00:45:32] The units, what are the units of volume?
[00:45:39] Yeah, it's whatever the units of r is cubed. So it was centimeters.
[00:45:46] Centimeters cubed per what time was it?
[00:45:53] The volume is decreasing at forty-eight hundred pi cubic centimeters per minute. How do I know it's decreasing?
[00:46:07] Because it is negative. When you write that out, you say the volume is decreasing at forty-eight hundred pi cubic centimeters per minute.
[00:46:30] You leave off the negative when you say it is decreasing.
[00:46:34] It is leaning up against the wall, so we have to draw that. Okay. Okay.
[00:47:01] Okay. The base of the ladder is being pulled away from the wall. Later on, it is going to look like that triangle.
[00:47:12] And eventually it looks like that and so on.
[00:47:18] A rate of three feet per second. All right. Let's put it to variables.
[00:47:28] What's changing here?
[00:47:34] Right. The height of the ladder is changing.
[00:47:39] Which we call that H.
[00:47:47] Let's go right now.
[00:47:55] Let's call that the base.
[00:48:01] Is the length of the ladder changing?
[00:48:05] No. So let's not put a Z here or whatever variable you want. Let's just plug in the number. This is twenty five feet.
[00:48:23] How fast is the well, let's go here. Being pulled away at three feet at a rate of three feet per second. What would that be?
[00:48:36] Sorry, d what?
[00:48:41] That's this distance right here is changing. That's db/dt. And is it getting larger? Is it increasing or decreasing?
[00:48:55] What's that? Increasing? Yes, increasing just three feet per second.
[00:49:03] So how fast is the top moving down the wall when the base of the ladder is twenty feet from the wall? So this is one moment in time.
[00:49:18] So what are we looking for?
[00:49:22] So we're looking for dh/dt.
[00:49:31] Except our answer is not going to be that value. I'll talk about that in a second.
[00:49:43] Okay, so what equation we come up with? How are h, how are b and h related at any point in time?
[00:49:53] We always have a triangle.
[00:49:57] What should we use?
[00:50:00] Area. Do we care about this area? No.
[00:50:06] Let's go with, this would be a what kind of triangle? Right triangle. Let's go with Pythagorean.
[00:50:12] h squared plus b squared equals twenty five squared.
[00:50:18] That's how b and h are related at any moment in time. Now we differentiate.
[00:50:29] Two h times dh/dt plus two b times db/dt.
[00:50:42] And derivative of twenty five squared is?
[00:50:47] Zero. Zero. It is just a constant.
[00:50:52] You could go twenty five, two times twenty five times the derivative of twenty five, which is zero. Don't do that.
[00:51:04] Okay, we want dh/dt.
[00:51:08] So let's solve for that.
[00:51:30] dh/dt is?
[00:51:47] We can simplify this a little bit.
[00:52:06] Okay.
[00:52:09] What was the length of the base? What were we given?
[00:52:19] So how when the base of the ladder is exactly twenty feet. So the one moment in time. So this is the rate only when this is twenty feet.
[00:52:31] db/dt is three.
[00:52:39] And how do we figure out height?
[00:52:46] We use this at that one moment in time.
[00:52:51] H squared is?
[00:52:55] Square minus b squared.
[00:53:07] B squared. What was the value of b?
[00:53:11] Twenty. Twenty.
[00:53:17] And h is? What is that?
[00:53:22] Twenty five.
[00:53:29] It is fifteen. It would have been easier if we just drew this as twenty, twenty five. What kind of triangle is this? Three, four, five.
[00:53:39] So it would be fifteen.
[00:53:44] Fifteen.
[00:53:51] And negative four.
[00:53:55] And it's the units of h divided by the units of time, which is? Feet per second.
[00:54:12] So how fast is the ladder moving down?
[00:54:18] Fast is like a speed, not a velocity. What we got here is a velocity.
[00:54:28] We've talked about difference in speed and velocity, right?
[00:54:36] So h, this distance is decreasing at four feet per second. That means exactly this point is moving that direction at four feet per second.
[00:54:49] Okay. dh/dt is not really how we want to phrase the answer. We want to know how fast this is moving down.
[00:55:00] Because it's moving down, this distance is actually.
[00:55:09] It's the rate we wanted, but we don't bother with that.
[00:55:16] Okay.
[00:55:19] Next, how fast is the angle between the ladder and the wall changing when the base is fifteen feet away?
[00:55:27] And where do I put the angle?
[00:55:32] We got two options.
[00:55:38] How fast the angle between the ladder and the wall. So that would be this angle.
[00:55:46] Before we do it.
[00:55:50] Is that angle growing or is it increasing or is it decreasing?
[00:55:57] As it goes down, this is getting a larger angle. So we're definitely better get a positive value.
[00:56:05] Okay. So anything involving theta, we use sine, cosine, or tangent. Which one should we use?
[00:56:25] Well, we're given db/dt. Okay. So use the one that involved B.
[00:56:36] Okay. So we want ideally use this.
[00:56:44] Okay. We could use tangent.
[00:56:54] But we'd have to know DHDT.
[00:56:59] Which we don't know at the moment. So let's use sine.
[00:57:06] Sine of theta is b over—
[00:57:11] And this is fixed. So put in the twenty-five.
[00:57:22] And we differentiate. We get cosine of theta times D theta, DT.
[00:57:33] This is simply one over twenty-five times the derivative of b, db/dt.
[00:57:50] Okay.
[00:58:22] When the base is fifteen feet away.
[00:58:27] First off, we know DB, DT, right? So that's, what was it?
[00:58:34] Three.
[00:58:37] Okay.
[00:58:48] So we need to know cosine of theta. At the one moment, the base is fifteen feet away.
[00:58:59] Cosine would be this length divided by this length.
[00:59:06] Adjacent over hypotenuse.
[00:59:09] So we need to figure out this length, which is,
[00:59:17] the Pythagorean theorem. This is what type of triangle?
[00:59:22] Three, four, five triangles. So we could do the math or know that that's twenty.
[00:59:32] So cosine of theta is twenty over twenty-five.
[00:59:39] Which is what? Four-fifths.
[00:59:50] Three over twenty-five times five over four.
[01:00:00] Which is three over twenty.
[01:00:11] Units of theta, always in this classes. Radians.
[01:00:18] Per time, which was, was it seconds?
[01:00:27] We get a positive value because that angle is getting larger.
[01:00:42] Okay. We're going to stop there.