>> Here's a problem involving the writing of a function. It says a rectangle is bounded by the X-axis and the semi-circle Y equals the square root of 25 minus X squared. Express the area of the rectangle as a function of X and determine the domain of the function. Well let's first talk about what it means for a rectangle to be bounded by the X-axis and a semi-circle. Here's the semi circle and of course the X-axis is along with the bottom. The rectangle could be configured in a number of ways. Just notice that the top of the rectangle is bounded by or the corners touch that semi-circle to cause the particular configuration that we have for the rectangle. And since we have a changing configuration of rectangle, we certainly have a changing idea for area as well. Now, the area depends upon the value of X. That is, as X changes value, as X changes along the X-axis the size of the rectangle changes. For example, for this X, here's the rectangle that corresponds to that particular X-value. And this kind of problem by the way represents the essence of the notion of function. We have a variable quantity X and that quantity is determining a different value of the area. Every time X changes, the area changes. Alright. To, for us to write the area in terms of that X-value we begin with the area formula. A is LW. Now understand that the length of our rectangle is really two values of X. Here's the value of X and here's another value of X or distance of X on the other side of the Y-axis. So, those two distances together really give us the length of the rectangle. The length then can be expressed as 2X. The width of the rectangle is the distance from the X-axis to the semi-circle. Well that's simply the Y, the vertical distance here but we know Y is the square root of 25 minus X squared. So, here's our area in terms of X and if we wanna write this in function notation, we could write it as A of X like this. Now, we're ready to determine the domain. The domain is determined according to the nature of the function at hand. This function only makes sense when the X-values are positive because only positive X-values will give us an area that is positive or 0. At the same time, we wanna make sure that our radicand is positive or 0 because that's the only time when the radical makes any sense. And the radicand is positive when X is less than or equal to 5. So our domain would be X is between 0 and 5. Well let's take this problem a little bit further. I'm a little curious about the largest rectangle possible. That is I'm curious about the largest area that we could possibly make here for one of the rectangles and the X that would cause that kind of situation to occur. Now, in order for us to investigate this, we can use a graphing calculator and to put this on to the graphing calculator we'll think of A of X as Y. We'll enter this equation on to the graphing calculator. I've already entered the equation here. Let's look at window settings. Now I've set the X minimum at 0, X maximum at 5 and notice the Y minimum at 0 and the Y max of 27. And the graph looks like this. Now, understand that this is a graph of actually the area because we are relating an X-value to a Y-value which is the area of the rectangle. And we want to maximize our area so let's go to the top of our graph using the trace button. I'm cursoring to the right now to get to the top of our graph. And I'm looking for the largest Y-value that comes up on display as I continue to cursor to the right. And I believe that largest value, I'm gonna go back this one here. Here it is 24.997, about 25 for the largest possible area and that occurs at rounding to the nearest 10th at an X of 3.5.