00:01
We're looking at maximizing function.
00:04
And we're given a practical problem, so we have a sheet of aluminium, and we know this is 20 inches wide.
00:13
And from here to here, it's 20.
00:18
And then what we do is we fold up the corners, so we get a shape like this, a square missing the top edge.
00:28
And now we know that its length is still 20.
00:32
However, we have three sides to it now.
00:38
So we know that these sides are equal, and we're going to call them both y.
00:42
And then we have this length here, we're going to call x.
00:48
And we know that we must have that x plus two lots of y must be our 20.
00:57
So that's our restriction.
01:00
Now we want to maximize the surface area, which is this area here.
01:08
Now we see this is given by x times y and so our surface area a is equal to times y.
01:25
That's just the red area and so this is a function we want to maximize because we want the maximum possible surface area.
01:35
So what we do is we're going to rearrange this equation in terms of x so we have x equals 20.
01:48
Minus 2y and we're going to substitute that into here so our equation for area just depends on one of our length and so what this becomes is a equal what x is 20 minus 2 y and then we can expand this 20 y minus 2 y square and then what we're going to do is we're going to factorize it.
02:24
So we take out a factor of 2y, and then we're left with 10 minus y.
02:36
And then we can see that we can draw a sketch over here...