00:01
Okay, what we're going to do now is a fun problem, actually.
00:05
So we're going to start with a long wire, and the wire is 12 feet long.
00:19
And we want to figure out where to cut the wire, where a portion of it will be x, and a portion of it will be 12 minus x.
00:33
And we want to figure out where to cut the wire to minimize the sum of the areas of the two shapes that we will make with these two portions.
00:58
And so this portion represented by x is going to make a circle with a radius of a radius of, radius of x over 2 pi and an area of pi r squared and the portion that is left over is now going to make a square with a side length of 12 minus x over 4 and an area of s squared which is 12 minus x over 4 the whole thing squared and so we want to minimize, we want to create a function of the sum of the areas.
01:51
And so the sum of the areas is going to be, and when i do that, i'm going to go ahead and rewrite these two areas, the area of the circle a little bit different and the area of the square of a little bit different.
02:03
So the area of the circle becomes x squared over 4 pi.
02:10
So i just squared the whole thing and reduced the pie.
02:12
And then plus the area of the square, which is 12 minus x squared over 16.
02:19
So this is the sum of the areas, and we want to minimize that sum.
02:26
And so that word minimize means that we're looking for extrema.
02:32
And so we know when we're looking for extrema, we need to take the derivative.
02:37
So we're going to go ahead and take the derivative in order to find the critical values.
02:41
And so this would be 2x over 4 pi plus 2 times 12 minus x times a negative 1 over 16...