00:01
We're asked to find the dimensions of a box here that has square bottom.
00:06
So therefore, the length and the width are the same.
00:09
I've labeled that as w both times.
00:13
As well, we were asked to find a box that has a volume of 12 feet cubed.
00:19
And we want to minimize the surface area of this, still having it.
00:26
And so if we want to minimize the surface area, we want to first off compute what the surface area has to be.
00:35
What is the formula for surface area? so the surface area can be thought of as just adding all the sides up.
00:44
So we have one of the surface areas.
00:46
We have this bottom box and this top box.
00:50
So two times w squared because we have a w squared contribution on the bottom and a w squared contribution on the top.
00:56
We then also have four side boxes with actually the same exact area, which would be width times height.
01:06
So we have four of width times heights.
01:11
So this is our surface area.
01:13
And this has to have, this has to be our volume because our volume is equal to w times w times because we have a square box again.
01:21
So we would get that 12 equals w square times h.
01:24
If we want to minimize this function, we should try to get this in terms of only one and so to do that, we can do this relatively easily by just dividing out by w squared.
01:36
And so we would get that h is equal to 12 divided by w squared.
01:40
And now what we're going to do is replace this h here and now get a function, particularly just of w.
01:47
So we get 2 w squared plus 4 times w times 12 over w squared.
01:56
More formally, we can write to clean this up a bit.
01:59
W squared plus 48 over w.
02:04
We now want to minimize this function.
02:07
We want to make this minimized.
02:09
So in order to find the minimization, we need to take a derivative, find the critical points, and go from there.
02:16
So let's find the critical points.
02:19
This will become 4w plus, well, actually this would be changed...