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The problem I'm going to look at here is that of taking a wire it's 8 meters long, and I want to cut it into two parts one parts going to form a circle and the other parts going to form a square and as usual in calculus we're looking at maximum and minimum here, and we want to either maximize or minimize the total area now well start off by letting X represent the circumference of the circle and then the remainder which I represent by 8 minus X will represent the perimeter of the square 2 simple formulas for circle C equals 2 PI R we could use PI D, but I like the fact that we can link up to the ass of the two formulas so what we're going to do is we'll put this X which represents the circumference of the circle in for C I want to substitute that into the area formula, so we get R equals x over 2 pi, and then I substitute that in for the R squared squaring that this is what I get and when I simplify I get 1/4 PI x squared now notice that I've tried to write this in the simplest possible way I'm going to take a derivative eventually, and it's I don't want to get into a product or a quotient rule here, so I easily have a constant in front of the x squared, so I've got a nice easy derivative secondly I'm going to take the remainder of this which is a minus X, and I'm going to use it to form the perimeter of the square now the perimeter is 4s area length times width or s squared and this time here's the perimeter that's where the X went for the circle circumference and if I divide by 4 I get some to skin 4s in the area formula now once again I want to make sure that I've got something that's easy to work with, so I squared the four on the bottom and I just removed that I've got a 1 over 16 out in front again hoping to make the derivative as simple as possible so the total area this is the area of the circle this is the area of the square when I take the derivative of this fairly simple it's just 2 times this X to the 1 and over here this is the chain rule but the derivative of what's inside is simply negative 1 but the derivative equal to 0 this negative sign I'm just putting out in front, and I'm sliding the 8 underneath there are several ways that you can simplify this I particularly like cross multiplying I just find it's so much easier if I just move one of the terms to the other side and just multiply the 2 pi times the numerator here in the 8 times the X on the other side that's what I get, and I move this term over because I want the X's on one side at this point I noticed I could divide through by 2 I could have done that earlier it just keeps the expression as simplified as possible factor out the X and then divide both sides by X plus 4 and that is the value for X I need form for the part that I'm cutting off for the circle now let's go back this was my expression for a total area, and you see I have a domain for X if X is 0 there's no circle I haven't allowed anything for the circle and if it's 8 it's entirely circled nothing left for the...
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