00:01
All right, so wish me luck on this problem.
00:02
It's been a while.
00:03
That's the vote of confidence speech i'm giving you at the beginning.
00:07
All right.
00:07
So we have this resort.
00:08
It's on a little island, just off the coast, i guess.
00:12
And there's a source of water, and they're going to run a pipeline to get water to the island.
00:18
Fresh water is probably the ocean, so we can't just pull it out of the water around it.
00:24
But anyways, the water source for the pipeline.
00:26
It's 20 miles down the shore.
00:28
And the island, it said, is 10 miles from the closest point on the shoreline.
00:34
So let me kind of draw this picture in for you.
00:39
This distance here is 10 miles.
00:44
That's what it's saying.
00:46
The distance all the way from this point to the water is 20 miles.
00:53
So we're going to break this up.
00:55
I'm going to call this little piece.
00:59
Like at some point it's going to go for a ways and then the pipeline is going to branch off and go into the water.
01:04
Water.
01:08
So i'm going to highlight that.
01:10
Well, i don't know what that point is, but what i do know is if i call this little section x, then this piece here is going to be 20 minus x because the, the water source is 20 miles down the shoreline from the closest point of the island.
01:33
That's how i'm reading this.
01:35
Water is 20 miles down.
01:36
Yeah, that's what it means.
01:37
I'm pretty confident that's what it means.
01:40
Okay.
01:41
So, here we go.
01:43
The total cost is going to be, here's our total cost.
01:48
Let's get a look at that.
01:50
The part that's on land is the 20 minus x miles, right? so the land part of it, the 20 minus x part of it is going to be, to lay water or lay pipe on land is $300 per month.
02:09
While.
02:10
So that times 300.
02:12
That's the dry land part.
02:15
And then the over the water part is going to be that green line.
02:22
Now, i don't know how long that is.
02:24
I don't know how long it is.
02:27
So it's going to be the green part.
02:30
I'm just going to put green times 500 because it's cost $500.
02:37
And we want that to be the smallest possible value.
02:40
Okay.
02:42
So look at it.
02:42
At that little triangle to find the green, we can use the pythagorean theorem.
02:47
It's going to be x squared plus 10 squared, and it's going to equal that the green part.
02:58
I'll put g for green part, g squared.
03:01
All right.
03:04
Now, this is useful because now if we take the square root of both sides, we now know that the green part, g, is going to equal the square root of x squared plus, i'm going to say 100, 10 squared is 100, which is good, because now i have something to put in for right here, for the green.
03:30
I have something to put there.
03:32
That's an x.
03:33
I can, you know, put the number there, or something there that has an x in it.
03:40
That's useful.
03:41
So i'm going to replace instead of just calling it the green.
03:44
Part, i'm going to replace it with that square root of 100, x squared plus 100.
03:50
So, x squared plus 100, x squared, not just x.
03:58
Don't forget the squared like i did.
04:00
Okay.
04:01
So, and that's times 500.
04:03
So here we go.
04:04
Now, if we want to minimize this, we got to figure out where it's going to have a horizontal tangent, so to speak.
04:11
Okay.
04:12
So what we got to do is take the derivative.
04:15
The derivative of the total cost, total cost prime, is going to be, so it's going to be negative 300.
04:32
That's what we get when we take the derivative of all this, plus, all right, so now we got to do some work.
04:41
So it's going to be 250 times x squared plus 100 to the negative one half.
04:53
And then we got a times this by the derivative inside the parentheses, which is just 2x.
05:00
All right.
05:01
You're wondering where that negative half come from.
05:03
So the square root of something is something raised to the half.
05:07
When i bring that down to the front, one half, and then i'll have, in parentheses, x squared plus 100, i got to subtract one from the exponent.
05:18
That's how i got the negative one half.
05:19
One half times the 500 is 250, 250...