00:01
For this problem, we want to find the maximum volume of a rectangular box with a square base, given that the sum of its height and base perimeter is 18 inches.
00:12
So first we want to write our objective function.
00:15
In this case, it will be the volume.
00:17
Now, since our box has a square base, then this is just x squared y, where x is the length of one side of the base, and y is the height.
00:31
Now since our objective function is written in terms of two variables, you have to rewrite it so that it will only be in terms of one variable.
00:40
We will use the fact that the sum of the height and the base perimeter is 18 inches.
00:46
So if the height and the base perimeter is 18, then this implies that y plus 4x is 18, so that y is equal to 18 minus 4x.
01:01
So, plugging it in our objective function, we have v equals x squared times 18 minus 4x, that's 18x squared minus 4x cubed.
01:17
Now that we have rewritten our volume function, we are now ready to differentiate v with respect to x.
01:26
So we have v prime, that's equal to 36x minus 12x squared, and then you will say, set this to 0 and 12 for x to get the critical number.
01:38
So 0 equals 36x minus 12x squared...