00:01
For this problem we are given that a box with square base and open top has a volume of 4 ,000 cubic centimeters.
00:08
We want to find the dimensions of this box that will minimize the amount of material used.
00:15
So if we are to illustrate it, this should be our box.
00:20
And if the box has a square base, then let's call that edge of the base, x.
00:27
So this will be x as well.
00:28
And its height, let's call it h.
00:32
So from here, we know v is equal to x squared times h, and this is equal to 4 ,000.
00:41
Now, the objective function here is the surface area of this box.
00:48
The objective function is the one that will be minimized or maximized in a given problem.
00:53
So for this, it's the surface area, since it's the one that will, describe the amount of material used.
01:03
The surface area here, since the top is open, would be x squared plus 4 times x times h.
01:14
Now we want our objective function to be written in terms of one variable, so we need to change this h here in terms of x using the volume formula.
01:25
So so since x squared times h equals 4 ,000, then this means that h is 4 ,000 all over x squared.
01:35
So our s a will be equal to x squared plus 4x times 4 ,000 all over x squared.
01:46
That's x squared plus 16 ,000 all over x.
01:54
And then we will differentiate our objective function...