00:01
Following a rectangular box.
00:03
So it's 40 by 20.
00:04
And basically in the upper left, the upper right, the lower left and lower right, we have these squares that we're cutting out of it.
00:11
And these squares have a dimension of x, right? and we're basically looking to find out what the volume of this box is.
00:20
Because if you can imagine, if we fold it up the sides, then like this would be our our base, right? and then we would have a box that looks like this, right, where our base is on the bottom.
00:40
All right, so our base would be here, and then our size x would be here.
00:48
Right.
00:49
So obviously, if we're trying to find the volume, we need the length and the width, right? because the equation for the volume of a box is length, times width, times height.
01:00
So let's write that v equals lwh.
01:04
All right, so let's start with the length.
01:07
So first, we know the entire length is 40, but we know that this part of that 40 is x, and we know this part is also x, just by looking at the diagram.
01:18
That means that the width of this base, if the whole thing is 40, we have two xs on the left and right, that means this width has to be 40, but we have to get rid of both of those xs.
01:31
In other words, we have to subtract.
01:34
So this width is 40 minus 2x.
01:39
Remember, i'm using 2x because i have an x on the left and on the right.
01:43
For the same like reasoning, our width is going to be 20 minus 2x.
01:51
Right? so this width here is going to be 20 minus 2x, right? so then this is going to be 40 minus 2x.
02:01
This is going to be 20 minus 2x.
02:04
And then our height, as we said before, are just x, because we're folding up those sides.
02:10
So like, that's going to make up the height of our box.
02:15
So then the volume enclosed is going to be equal to length, which is 40 minus 2x, right, times width, which is 20 minus 2x, times height, which is x.
02:31
So this is an expression for our value.
02:35
Go and multiply things in and like all that stuff.
02:38
But we can really just leave it like this.
02:41
This is a simplified factored form.
02:45
So now that we have our volume, the like the next question is what is the domain of x, right? so we're looking for the domain now.
02:56
Now, because we're dealing with a box, that means that our volume has to be positive because we can't have a like a, a, like negative volume and all our dimensions also have to be positive.
03:15
So if our, if our volume is greater than zero, that means we're going to have, let's just do this in different color.
03:27
All right.
03:27
That means that, oops, okay, so that means our h is greater than zero as well.
03:34
H greater than zero.
03:35
Okay.
03:37
So if v is greater than zero, that means we have 40 minus x, 20 minus 2x, times x has to be greater than zero.
03:50
We also have that each individual thing also has to be greater than zero as well.
03:56
The length, the width also has to be greater than zero, right? and the height, x greater than zero.
04:06
So we basically have to look at all four of these situations to make sure that all like a four of these are being taken into consideration.
04:15
All right, so let's actually start with the length and the width and the height, because these are the easier ones...