 Alright, let's look at another related rates problem. And the biggest challenge to setting up a related rates problem is to write down a relationship between the variables. And often the biggest challenge there is identifying what's actually a variable. So for example here, you have a 20 foot ladder leaning up against a wall, the base of the ladder is pulled away from the wall at 5 feet per second, the top of the ladder maintains contact with the wall. How rapidly is the top of the ladder moving when the base is 8 feet from the wall? So useful any time we have the option of doing so, let's sketch a picture of the situation described. So here is my picture of great artistic merit showing a ladder up against a wall. And probably the most useful question to ask here is, if something changes, we should represent it by a variable. So we want to know what changes and what stays the same. So here's an important possibility. What is the height of the top point of the ladder change? Well, let's draw a couple of pictures at the very beginning. The top of the ladder is up here someplace. And as I pull the base of the ladder away from the wall, the top maintains contact with the wall. So I'm going to pull this away from the wall a little bit, like that. Let's try that again, I'm going to pull it away. I'm going to pull it, pull, move, move, there we go. Alright, so there I've dropped the ladder down. And I'm going to pull that a little bit further away. I'm going to give it another pull, pull it away a little further, and it's there. And so notice that the top of the ladder was up here, but it's dropped down to here, which means that the height changes so it should be represented by a variable. How about the length of the ladder? So at the beginning, I have a ladder that's this long. And as I pull the base away from the wall, I have a ladder that is still the same length. And if I keep pulling it away from the wall, the length of that ladder is still 20 feet. And since the length of the ladder doesn't change, we can use the actual length of 20 feet. Finally, how about the distance of the base of the ladder from the wall? Well, part of the thing that will make this confusing is that we're interested in what happens when the base is 8 feet from the wall. And if we read this as a constant, we're going to have a very difficult time with the problem. So let's analyze this. Does the distance of the base change? So right now, the base is so far from the wall, but if I move the base of the ladder away from the wall, maybe to here or maybe to here, the base gets farther and farther away from the wall, the base is a variable. And so it should be represented by a variable. And so that says that I have one thing that's constant, the length of the ladder, and I have two things that are variable, the height of the ladder and the length of the base. So I want to represent those as variables and the length is a constant. Now the information given tells us that the length of the base is changing at some rate by feet per second. And so that's a rate of change with respect to time. So I know that the derivative dB dt is equal to 5. I am interested in knowing how rapidly the top of the ladder is moving. So that's going to be a rate of change of H, again, presumably with respect to time, because that is what our units are given in. So I want to find dH dt, and in particular, I'm interested in that value when the base is 8 feet from the wall, when B is equal to 8. Now remember, B is a variable, so I'm not going to substitute this value in until the very end of the process. So first off, I can look for any relationship whatsoever between the variables that I have. Well, it's B, that's H. If I knew the area of this triangle, I might be able to express it, but I don't. So how about the Pythagorean theorem? That's a good one to fall back on. So I have a right triangle there, so I know by the Pythagorean theorem, B squared plus 8 squared equals 20 squared, and there is some relationship between the variables. Now again, I'm looking for the derivative with respect to time, so I'll differentiate with respect to t. And a little bit of analysis goes a long way. So over on the left-hand side, I have 20 squared is a constant, so if I differentiate that, I get 0. Over on the right-hand side, I have a sum, so if I differentiate a sum, I get the sum of the derivatives. And again, the thing to remember is chain-roll, chain-roll, and chain-roll. If you're not applying the chain-roll, there's a good chance you're missing something. So let's see, I have two functions here, they're both squaring functions, so to make things easier, I'll ignore everything except for that last function. I have the derivative of 8 squared, and so that's going to be 2 times the derivative. So there's a chain-roll and the kindergarten-roll, put everything back where you found it. And so I have 2b times the derivative of b, 2h times the derivative of h, as my relationship between the derivatives. Now if I know three of these things, I can figure out the fourth one. So let's see, what do I know? Well I know I'm interested in a place where the base is 8 feet away from the wall, so I know b is 8. I know that db dt is 5, the base is being pulled away from the wall at 5 feet per second. So I know b, I know db dt, I have two things I am looking for, I am interested in how rapidly the top of the ladder is moving, so I want to know this dh dt. So I need to know h. Well if I know that b is 8, then I can use this original relationship between b and h to figure out what h is. So if b is 8, I know this is 20, and so that tells me h after all the dust settles is square root 336. So that's just substituting my value of b into this equation here and solving for h. And if I make those substitutions, I end up with this expression which has just one thing I don't know, dh dt. And I can solve for dh dt is minus 8 over square root 336. Always look for what the units are. This is the rate of change of h with respect to time. This is a change in h is h is the height, that's got to be something in feet, with respect to time, that's a change in time and the only time units we have are seconds. So this value dh dt must be an amount that's expressed in feet per second. A useful thing to do if we happen to be able to do it in the particular case in question is to consider the fact that dh dt is computed to be a negative amount. And what that says is that h is decreasing. And if I actually do pull the base of the wall, base of the ladder away from the wall, I would expect h to be decreasing. So here's a nice reality check. The final answer that I get indicates that h is decreasing and that's exactly what I would expect in this physical situation. So there's many possible places that this computation could have gone wrong if I'd gotten the derivative wrong, if I'd gotten the values of h wrong, if I'd gotten any of a number of other things wrong. But at the very least, I'm getting an answer that is heading in the right direction. So that's a good check to make anytime you finish a problem like this.