>> Consider this problem. Suppose an object is propelled at a velocity of 48 feet per second on an angle of 45 degrees. So here is that object propelled at an angle of 45 degrees at a rate of 48 feet per second. Now we know that this object is traveling. We can think of it as traveling horizontally and vertically at the same time. It's on an angle right here but it's generally moving to the left than it's moving up. Now this is nothing more than separating the vector into its components. Now from trigonometry, we know that that's accomplished using trig ratios. And the horizontal rate associated with this situation is found by using the cosine of 45 degrees because cosine of 45 degrees is the adjacent side over the hypothenuse so that it would be R over 48. And the vertical rate are subscript V, I can indicate it as--is found by using the sine ratio, the sine of 45 degrees is R over 48. Okay, let's do that and let's talk about--and let's introduce the notion of time here at the same--in conjunction with this. Alright, horizontal rate is cosine 45 degrees which is R over 48. So R then is 48 cosine 45 degrees but cosine 45 degrees is 1 over root 2. If I rationalize here I'll accomplish a nice little cancellation and I'll remove the fraction. Here is what I mean. This becomes root 2 over 2. I get the cancellation here. So the rate then is 24 root 2. Now, I can associate this with another variable, time, by using the distance formula. You know, we're talking about a horizontal rate here and so the distance formula--distance is rate times time comes into play. Now the rate is 24 root 2 times time. Now if we're talking about a horizontal idea, horizontal distance, that's what X is all about when we talk about the X, Y plane. So X then is 24 root 2 times T. Now consider vertical rate. Vertical rate is sine 45 degrees which is R over 48. So R then is 48 sine 45 degrees. Sine of 45 degrees is also 1 over root 2. We get the same rationalization, the same cancellation. The rate then is 24 root 2. So now, this is a vertical rate at the moment that this object is propelled on that angle at that certain velocity. But immediately, gravity will begin to take its effect. And that object will slow down and then descend back to the earth. Now the formula relating that idea that is height in relationship with time and this initial velocity is this. It's H equals initial velocity times time minus 16T squared. And by the way this is derived from one of Newton's laws. So initial velocity, well that initial velocity is this vertical rate, you see, that we talked about. That's 24 root 2. And now just bring down the other element. Now since we're talking about a height here, a vertical distance, you see, that's just the Y idea. So a vertical distance is this expression. Now we have 2 equations and both of them, this X equation and this Y equation both involve time. So we have a variable X, a variable Y, and a variable T. Now what are we gonna do with these? Well, it turns out that the 2 equations that we're talking about for X and Y are called "parametric equations." And the variable in both of those equations is T. And so T is called the "parameter" involved in this situation. Okay, let's see how we can analyze this. The parametric equations are these two equations. Now one way to deal with it, one way to analyze it is like this. We can make a table of values involving the 3 variables. So we can let T take on various values and we could plug those various values of T into the separate equations. So if T is 0 then add all the terms go out and X is 0, Y is 0. If T is 1 then the X equation becomes 34. The Y equation becomes 18, you see. So when T is 1, X is 34, Y is 18. This means that after a time of 1 second the horizontal distance is 34. The vertical distance is 18, you see. And then for our T of 2, X turns out to be 68 and Y turns out to be 4. Now we could actually plot the coordinates for the Xs in Ys, you see, on a coordinate plane and we could attach to that--to that graph information about time. We could say at this point, T equals 0. At this point, T equals 1. At this point, T equals 2. Now notice that in thinking about the passage of time though, we create a certain linearity involved in this whole study that time begins at 0 and there is a passage of time and then there is the last moment for our study in this problem. So there is a certain orientation to the graph that is the graph has a certain flow to it in a certain direction and that's kind of an important idea. By the way that graph is called a plane curve, a plane curve in this situation. Now another way to deal with all of these and in such a graph like this we could read the graph and we would have all three bits of information at one time. But another way to deal with it is to take the 2 parametric equations and go through a process of eliminating the parameter. That is we could actually eliminate T. And to do that we take 1 equation or 1 technic for doing that. I'll show you another one a little bit later. But 1 technic is to solve for T and 1 or the other equation and then substitute that information into the other equation. In this circumstance it's easy to solve for T in the first equation. So that's the first thing we would do. We would find T to be X over 24 root 2. Now take this information about T and put it into the second equation. So the second equation becomes Y equals, you see, and instead of the T here we put our information from the other equation. And instead of a T here we put the information from the other equation. And when we simplify a bit, we find Y to be X minus X squared over 72. Now it turns out--