 Hi everyone, let's take a look at an example of a related rate problem. In this one we have a boat that is being pulled to a dock at the rate of 2 feet per second. The pulley is 8 feet from the water and the rope is tied to the boat at 1 foot above the water. How fast is the boat approaching the dock when the boat is 20 feet away from the dock? So if we were to try to draw ourselves a picture first and identify all the rates that were given and those that we're trying to find, then we can set up an equation to solve and remember related rate problems are solved through implicit differentiating each variable with respect to t, the time. So let's draw ourselves a picture first. So suppose here's the dock and perhaps this is the pulley system and let's draw ourselves a boat and we are told that the rope is tied to the boat at 1 foot above the water. So this distance would be 1. Let's assume that the rope is right here and we know that the pulley is 8 feet from the water. So we're basically going to make a right triangle out of this. So if this height here is 1, that means if we make a right triangle out of this, this height is going to be 7. And we are told that the boat is being pulled to the dock at a rate of 2 feet per second. So let's think about what that's telling us. Remember it's the pulley and the rope that is pulling the boat into the dock. So really if we identify this side over here as perhaps z, let's call this horizontal distance down here x that represents how far horizontally the boat is from the dock. We could express the fact that we are given dz dt is the 2 feet per second. Now what we're trying to find is how fast the boat is approaching the dock. So we're trying to find dx dt at the specific point in time when x is 20 feet. So what we could do is knowing that at that particular point in time when x is 20, we can apply the Pythagorean theorem to find out what z will be at that point in time. So let's go ahead and do that because we're going to need that later anyway. So we might as well do it right here. And when you solve that out, you get that z is the square root of 449. You can leave it that way. You can simplify it if you wish. You could express it as a decimal. Although if you are going to express it as a decimal, be sure to store that value in your calculator for use later. So maybe it's just as easy to leave it as square root of 449. So now we need an equation that's going to tie everything together that we can do the implicit differentiation on. So given we have a right triangle, and it sounds like a Pythagorean theorem one, we're going to go ahead and set it up as a right triangle with the Pythagorean theorem. So that means we have x squared plus 7 squared equals z squared. And we go ahead and do our implicit differentiation on that, differentiating each variable with respect to time. Derivative of 7, of course, is simply 0, and derivative of z square is 2z dz dt. So remember we do not substitute any values in until after we've done our implicit differentiation. So now we can go ahead and substitute in those values that we know. So at the particular point in time at which we're trying to figure out how fast the boat is approaching the dock, we know x is 20. dx dt is that which we're trying to find. Remember we figured out z at that point in time to be the square root of 449. And dz dt we know at that point is 2. So if we go ahead and solve for dx dt, that gives us square root of 449 over 10. We could express that as a decimal if you wish. So that is approximately 2.119. So our conclusion therefore is that when the boat is 20 feet from the dock, the boat is approaching the dock at a rate of approximately 2.119 feet per second.