00:01
Okay, so here's our picture.
00:03
And so the farmer starts at his house, you know, in the morning, goes to the river, get some water, and then brings it down to the farm.
00:16
Okay, now he could pick.
00:17
He didn't have to go to that specific point p.
00:20
He just has to hit the river somewhere.
00:24
So we want to minimize his walking distance.
00:27
Okay, so here's a couple things i'm going to do.
00:30
I'm going to call this distance here x, which would make this distance here 6 minus x.
00:44
Okay? so the distance that the farmer is going to walk, if we use the pythagorean theorem, we know that from a house to the water is going to be 2 squared plus x squared.
01:12
This is 4 plus x squared now the other side so 5 squared is 25 oh this is under that's under square root this one here 25 plus and then when you square the 6 minus x you get 36 minus 12x plus x squared so his walking he's going to walk 4 plus x squared plus 61 minus 12 x plus x squared so all right so we want to we want to know that we want to find the the we want to minimize this basically okay what am this is kind of a messed up formula but okay so w prime take the derivative one half i'm going to do this quicker.
02:56
2 square root of 4 plus x squared.
03:01
And then the derivative inside is just 2x.
03:04
So 2x is going to be on top plus.
03:10
And we've got the same thing over here.
03:13
2 square root of 61 minus 12x plus x squared.
03:20
And then on top it'll be 2x minus 12.
03:23
All right.
03:26
Now we've got to set this thing equal to zero.
03:29
It has to equal zero.
03:31
All right.
03:31
Now, the denominator equaling zero doesn't do us any good.
03:37
But the numerator equaling zero, that's going to be important.
03:43
That's going to be important.
03:44
Okay.
03:45
The problem is we don't have a common denominator.
03:48
Terrific.
03:49
Well, we've got to get a common denominator.
03:53
Great.
03:53
Let's simplify this down a little bit.
03:55
These twos can reduce out.
03:58
If i factor a two out and write x minus six, these twos will reduce out.
04:08
So maybe that'll kind of make life a little easier for us, i hope.
04:21
4 plus x squared, x minus six on top here.
04:32
61 minus 12x plus x squared.
04:37
All right.
04:39
Gosh, it would be nice if we could just set those numerators equal to zero, but we can't.
04:44
Oh, this is going to be bad.
05:01
Okay, so i'm going to multiply through by these.
05:03
I'm going to set it equal to zero.
05:06
So when i multiply through by the denominators, i'll get zero equals x times the square root of 61 minus, 12x plus x squared plus x squared plus x minus six times square root of 4 plus x squared.
05:35
All right.
05:37
So negative x square root of 61 minus 12x plus x squared...