00:01
Here we want to make an open box and we're going to start with a sheet of cardboard, which is 12 by 16.
00:12
We're going to cut out the corners and we'll cut out squares of length x.
00:19
And we want to know how much should we cut out so that we can fold up the sides to make the largest volume possible.
00:26
So to start with, i'm just going to write an equation for the volume.
00:29
I'm going to write it in terms of whatever variables i can find.
00:32
So say length times width times height.
00:36
Now let's make each of these variables a little more simple, and maybe we can write it in terms of all one variable.
00:43
That's our real goal here.
00:44
That way we can take the derivative and set it equal to zero.
00:48
So if i consider what the length is, well, it started off with 12, but then we cut out x twice.
00:54
So really the length is 12 minus 2x.
00:59
And i think the same thing can be said about the width is 16, and we take away 2x.
01:05
Then we fold up the sides and we get a height of x.
01:08
So the volume is those three things multiplied together.
01:12
Since we have it in terms of one variable, we could take the derivative and set it equal to zero.
01:17
But we'd have to use the product rule a few times, and i don't want to do that...