 So let's take a look at finding other objective functions. Again, this is not really a calculus topic. This is a algebra topic, and if you have a good way of constructing functions that works for you, go ahead and use it. If this is something that you have a problem with, that it's something you have difficulties with, this is something that might help you find those functions. So here's another example. Let's take a 30 centimeter length of wire. We're going to cut it into two pieces. We're going to make a square out of one and a circle out of the other, and what we'd like to do is find a function of the total area of the two pieces as a function of the length of one of the pieces. So again, it helps to consider what is a constant and what is actually a variable. So the piece that we're going to form into a square, it's a variable. We're going to cut the wire into two pieces, and one of them is going to become the square. So we'll let x be the length of the piece that's formed into a square. And as before, it helps to form a table. So we want the total area, which means that we want to find is the area of the square and the area of the circle as a function of the length of one piece. And so I've constructed a table, and here's a really, really, really important idea to keep in mind. Paper is cheap. You cannot go wrong by taking up too much space. Paper is cheap. And the important idea here is that the more you have to keep in your head while you solve a problem, the less brain space you have to devote to actually solving the problem. So what you can do is if you can shift a lot of that onto cheap paper, you can free up brain space that allows you to actually solve the problem. The more you write down, the less you have to remember. So let's see. I'm going to find the total area, well, that's the area of a square plus the area of a circle, based off of how big the piece that's formed into a square is going to be. So if I want to find the objective function, let's pick an arbitrary value of x and see what happens. So I'm going to pick a number, how about square root of 17 over 35 plus pi over 11. Well, let's pick an easy number. How about 12? 12 is a nice, even number. It's a dozen. And if I take a piece of 12 centimeters and form it into a square, what's the area? Well, maybe the answer is not obvious, but if I want to find the area of a square, I go through my geometry formulas and remember that if I want to find the area of a square, I just need to know the length of one side of the square. So I need to know the side of the square, and I'll fill that in as one of the pieces of information I need to know. So I'm going to take this wire of length 12 centimeters and form it into a square, and since all four sides of a square have the same length, then that 12 centimeter segment of wire is going to have sides of 12 divided by 4 centimeters. And here's a useful idea. Don't do the arithmetic. It's going to make the algebra harder to recover later on. We can figure out what the side of the square is, but if we don't do the arithmetic, our lives are going to be much easier. Well, now, once I know the side of a square, I know how to find the area of the square, and that's just going to be the square of that side, 12 over 4 quantity squared. And again, your life will be much easier if you don't simplify the expression. How about that circle? Well, if I want to find the area of a circle, I need to find the radius of the circle. Well, so let's see. I've already used 12 centimeters of the wire to form a square. So how do I find the radius? Well, I'm going to take the rest of it and form it into a circle. So maybe it'll be helpful if I find the remaining length. So the remaining length is going to be, well, if I've used up 12, then the amount that I have left, I was starting with 30 centimeters. The amount that I have left is going to be 30 minus 12. And again, it helps if you don't simplify the expression. Now, the amount that we formed into a circle, the circumference is going to be 30 minus 12 centimeters. And since I have this nice relationship between circumference and radius, I can substitute my circumference 30 minus 12 into this equation and solve for the radius 30 minus 12 over 2 pi. And then once I have the radius, I can find the area of the circle, pi times radius squared. So if I take a piece of length 12 centimeters and form it into a square with the remainder going into a circle, the area of the square, 12 over 4 squared, the area of the circle, pi times 30 minus 12 over 2 pi quantity squared. Now, if you're really good and you're paying very close attention, you can actually write down the function at this point. But again, that requires a lot of brain space. So let's take advantage of the fact that paper is cheap and write down another sample and take another arbitrary length, how about 20? And the nice thing about this is having done this problem once, we actually have a pretty good insight into how to fill in the rest of the table, because 20 centimeters, I need the side of the square. That's going to be 20 over 4. I need the area of the square. I'm going to square 20 over 4 squared. I need the remaining length. That's going to be 30 minus the amount that I've used. I need the radius. So remember, the remaining length became the circumference of a circle, so I can divide that by 2 pi to get the radius, and then the area of the circle pi times the radius squared. And there's my area of the circle, area of the square, and add them together to find the total area. So now we're ready to take that leap. What if our arbitrary length is x? And again, we play the game. One of these things is not like the other. So everything here in this first column over 4, and the number in the first column is the numerator. So our first column is going to be x over 4. Let's take a look at the next column. The next column is just whatever number is in here squared. So our next column, x over 4 squared. How about our third column? Remaining length. Everything has a 30 minus, and the number that's being subtracted is the same as what's in the first column, 30 minus 12, 30 minus 20. That says this should be 30 minus x. Next column. Numerator is whatever was here. Denominator is always 2 pi. So our next column, numerator is this. Denominator is 2 pi. Last column. Everything has a pi, and whatever was in the previous column squared. So our last column, pi times 30 minus x over 2 pi quantity squared. And now I have the area of the square, the area of the circle. And so if I want to find the total area if the length formed into the square is x, that's just going to be the area of the square plus the area of the circle. And there's my function.