00:01
Okay, we have a square piece of cardboard or tin or whatever, whatever the material we have, but we're cutting x by x.
00:10
So then what we see is something like this, right? a box with a height of x, and this is 9 minus 2x, and this is 9 minus 2x, right? you see that from the shape.
00:29
So the volume of this box will be x times 9 minus 2x times 9 minus 2x, which is squared.
00:37
So i'm gonna multiply this out, and i'm gonna get 4x cubed minus 36x squared plus 81x, okay? to maximize this, we need to take derivatives.
00:57
12x squared minus 72x plus 81, set it to zero.
01:03
It never becomes undefined.
01:05
So then i get x equals 3 over 2, and x equals 9 over 2, okay? which one is max, which one is minimum? well, let's use the second derivative test.
01:18
Let's take the derivative of the volume second times.
01:21
We get 24x minus 72...