Calculus: The Ladder Problem

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As you can tell by our wonderful set back here, it's time for the famous falling ladder problem. So let me tell you about this problem. We have a ladder that's leaning against a wall, just like this is. And what happens is, for some scary reason, the bottom of the ladder starts to slide out and the top of the ladder starts to fall. And of course, people are scared by that, because that means lawsuit city if you get hit. So now here's the question. Suppose that the ladder is 13 feet long. And as the ladder is sliding out, we know how fast the bottom is moving out from the ladder. The question is, how fast is the very top of the ladder shooting down? To try to give you a sense, first of all, of exactly what's going on, I thought I would actually enact this for you. Now I'm not going to use the big ladder, because my insurance won't cover it. But my insurance will cover the use of this ladder.
So let me set up a little wall here for your and let's try to run this experiment. So the ladder you see is 13 feet long and is leaning against this wall. And what happens is, for some reason, someone pulls the ladder out. And what you're going to see is, the ladder is now going to shoot down. Now the top of the ladder will always stay against the wall as it goes down and the bottom of the ladder, you see, is going to shoot out this way. Let me do this for you, right now, as it actually would happen. So watch this, it's really fast. There it goes. Now I want to do that again for you in slow motion so you can really see what's going on here. So what happens is the bottom starts to slide out and notice that the top is slipping down. And the question is, if we know how fast the bottom is sliding out, how fast is the top slipping down? Maybe you think, in fact, it's the same rate, because it's the same ladder. So whatever the bottom is doing, the top must do the same thing. Well, let's see and maybe we'll discover your intuition is correct.
So that's the motion. I know how fast the bottom is sliding out. In fact, the bottom is sliding out at the very instant that this distance, away from the ladder to the wall, is 12 feet. Remember the ladder itself is actually 13 feet. At that very instant, where we're 12 feet away, then we're told that the bottom of the ladder is moving at a rate of 6 feet per second, so it's really just moving really fast - 6 feet per second. And the question is, at that very instant when the ladder, which is now located - it's base is located - 12 feet away from the wall and moving at a rate of 6 feet per second, how fast at that instant is the top falling downward? Let's see if we can actually model this and use calculus to figure out the answer.
So the first thing we have to do is remember our method of solving all of life's problems. So what's the first thing we should do? Well, the first thing we should do is understand what is being asked. And in a problem like this, this problem really cries out for a picture. The first thing you should do is try to draw the picture. Of course, I'm able to enact it right here for you live, but if you don't just happen to have a ladder, then you may want to actually just draw a picture of this. So here's the wall. Here's the floor. Everything, by the way, is straight, 90 degrees. Everything meets perfectly. It's not a cheapo house. And then we have the ladder leaning up against the wall. And the ladder, we know, is 13 feet. So we should write that in there. The ladder is 13 feet. So it's a 13-foot ladder and that length doesn't change. So it starts there and then it starts to slide down, just like that. So it's up here and it starts to slide down.
Well, now what are we trying to figure out? We're trying to find out how the top of the ladder is moving. At what rate is that moving downward? You can see, the top really does slide down, and at what rate? Now how am I going to write that down? Well, I could write down in words. There would be a lot of words. We want to find out how fast the top of the ladder is falling at the very instant when the bottom of the ladder is 12 feet away from the wall and moving at a rate of 6 feet per second. So you can write all that down, but instead, I'm going to immediately introduce symbols. Part of the power of mathematics is that it allows us to say things in a very succinct manner.
So to figure out how this distance is changing, let me give that distance a name. Now we could call it x. We could call it y. I'm going to call it y because I think of it as the y direction. So let me call y the length from the top of the ladder to the floor. That distance, I'm going to call that y. So y is this.
Now what am I trying to figure out? I'm trying to find out how that distance is changing, because for this ladder to be falling down, that means that this y distance is actually shrinking. So what I want to find out is how y - how that distance - is changing. That's the question. And so therefore, I would say, well, I want to find d what, d what? I want to find . I want to find out how the height is changing with respect to time. That will tell me how fast that top is falling down. And how do I want to find that? I want to find that when the base of the ladder is located 12 feet away from the wall. So at the very second, when that ladder is 12 feet away from the wall - right now! - how fast was it moving down?
So when the base of the ladder is 12 feet from that wall - let me actually give a name for the distance from the wall to the base of the ladder. I'll call that x. Instead of writing all that out, we'll call that x. And so therefore, I say when x = 12 feet. So question is to find at the moment when x = 12 feet, where this is x and that's y. Well that's what we're being asked here. How do we proceed next? Now we determine what we know.
What is that we do know? Well what we do know is we know how fast x is changing, because we were told how fast the bottom of the ladder is sliding out. See the bottom of the ladder is sliding out like this. That's what's causing the top of the ladder to fall. I'll do it again for you - enact it - and you can watch and see that what's happening is the bottom is sliding out and the top is falling out. And we know that speed. We were told that speed. Someone computed that for us. So what we know is I know how fast x is changing, with respect to time. So I know that - that's the rate of change of this length with respect to time - is equal to what? Well we were told it's 6 feet per second. So this is what I want and this is what I know.
And our final step in solving all of life's problems is to find a connection - a relationship - that links what I know and what I want together. So this is really where the creativity comes in. I think both of these steps, by the way, are extremely difficult in and of themselves. But then when you come to step three, this is where all the thinking happens. You need some new idea.
You'll notice in these problems, by the way, there's no set rule or list of things to do like in max and minimum problems where you take a derivative, set it equal to zero and solve. There's no algorithm like that here. You really have to think about the problem, model it, set it up, give names to variables, figure out what you know, figure out what you want, and then figure out the connection. It's really hard.
Let's think about this now. What's a connection that links x and y together? Well, if you look at this picture, you may recognize the fact that we have a right triangle here. So in fact, we know the Pythagorean theorem is going to hold, and that's a relationship that will link x and y together. So the connection is, we note that x^2 + y^2 equals this length - but that's fixed; that's the length of the ladder - 13^2. So notice, that's a connection that links y to x. I want to find out how y is changing with respect to time, but I know how x is changing with respect to time. So if I take this fact and differentiate it with respect to time, then I will have all the characters in sight that I'm trying to study. So in fact, I think now we're about to make some progress into this problem. Let's take a look and see what we do.
I now want to differentiate, with respect to time, x^2 + y^2 = 13^2. And as always, we'll have to use implicit differentiation since these are not t's. They just depend implicitly on t. So what's the derivative of x^2? The derivative of x^2 with respect to t requires a little chain rule. It's blop squared, so the derivative of it is 2 blop - so 2x - multiplied by the derivative - I just peeled off a 2 - the derivative of x. What is the derivative of x with respect to t? The derivative of x with respect to t has a name. It's called ,
So that's the derivative of the first piece. Now let's take the derivative of this. Here I see this as blop squared since there are no t's there. The derivative of blop squared is 2 blop multiplied by the derivative of the blop. I just peeled off that 2. The derivative of y with respect to t has a name. It's called . And that equals - and this is really easy. That's a constant, so its derivative is zero.
So now I took this connection and converted it, using derivatives - taking the derivative - into a connection that links rates together. And look, this is the rate that I'm trying to find. This is a rate that I already know. So in fact, I could put the pieces together and figure out what is. So let's insert all the information now.
Well I know that x has to be 12 feet at the very instance I'm trying to find this. So I know that at that moment, x is 12. So I could put it here, 2 x 12. At that instant, I know that is 6. And I have to add to that 2 times y. Uh oh, what's y now? Well, gee, I don't know. y was never given to us. What is what? Well, I guess is how high the top of the ladder is from the floor at the very instant when this is located 12 feet away. So at a particular moment in time, like that time right there, how high is that? Well, I know this is 13. I know this is 12. So I guess I could actually us the Pythagorean theorem to find out what y is. So I could go off and do a little separate calculation if I wanted to. But notice that at this very instant, what I have here is something like this. I know this is 12 at that instant. The ladder remains 13. And this is y at that instant, and I want to find that. So you can use the Pythagorean theorem or recognize that this is actually a 5/12/13 right triangle. So in fact, y is five.
If you didn't know that, by the way, just work it out. If you take 13 and square it, you get 169. If you take 12^2, you get 144. When you take 169 and subtract off the 12^2, you see 25 and so therefore, y must be the square root of 25, which is 5.
Anyway, we see that y, at this moment, is actually 5. And is the thing I don't know. I'm trying to find that. And all that equals zero. Well now, everything is known except . I can solve this for , and so what do I see? I see that equals - I'll bring all this stuff to the other side. So I see . And the twos cancel, and so what am I left with? I'm sorry. This is a typo here. This should be . Sorry about that. therefore equals - and what's 6 x 12? I think that's . And units here are feet per second. What does that equal roughly, by the way? If you take 72 and divide it by 5, we see this equals -14.4 feet per second.
So what does that mean? Well, it's really falling fast. The base of the ladder is moving out at a rate of 6 feet per second, yet the top is just zooming down at a very, very fast speed. And maybe you noticed when we did the example at the beginning, that top really falls down very quickly. You can go back and you can watch it and see it really, really falls down much quicker than you may think.
The last thing I just want to mention here before we go on to the next question is that negative sign. What does that negative sign mean? Well, it means that the velocity is negative - that in fact, the velocity is actually decreasing. And so in fact, what we're seeking here is that this rate is actually in a downward direction. So in fact, when you have a vertical motion of this sort, the sign tells me whether I am going up or going down. So the fact that I have a negative answer here tells me that the ladder is falling down. Does that sound right? It certainly does, because in fact, the ladder is falling down. If we did all this work and we ended up with a positive 14.4, if you would think about it, you could convince yourself that something had to be wrong. Either you forgot a negative sign or you made some larger mistake, because this thing is falling downward and so therefore, the answer needs to be negative. This distance is shrinking and so the rate should be negative. So that's what the negative sign means, and that's the answer - 14.4 - pretty fast. This is why you don't want to be standing under a ladder when it starts to fall. Okay, up next, we'll take a look at another classic calculus caper application. See you there. Bye.