00:01
Here's a diagram that you see in your book.
00:03
So this is the house and this is the cable box.
00:06
So we're trying to find a point along the road so that we can minimize the total cost of the cable lines.
00:15
So this blue line is the cable line that's along the road.
00:21
Okay, so this blue line costs $500 per kilometer.
00:29
And then the red line is the cable line that's...
00:33
Away from the road and that one costs $600 per kilometer.
00:41
And these distances are given in terms of kilometers as well.
00:45
Okay, so we can see that for the blue part, the total cost is going to be the distance, which is five minus x times the cost per kilometer.
00:55
So let's start by writing a cost function.
01:02
That's going to be, so let's start with the blue one, $500 per kilometer times five minus x.
01:10
Minus x kilometers.
01:12
And now the red cable, that's $600 per kilometer.
01:18
And the distance there, well, that's just a hypotenuse of this triangle.
01:22
So the total distance for that one is going to be the square root of 2 squared plus x squared.
01:34
So now we have a cost function and we wish to minimize this.
01:40
So then that means we need to go ahead and find the derivative.
01:44
Instead of equal to 0.
01:46
Okay, so we get this as equal to negative 500 plus 600 times.
01:55
So this is going to be one half of 4 plus x squared to the power of negative 1 half times 2x.
02:09
Okay, so let's simplify that a little bit.
02:11
This happened, the 2 will cancel.
02:14
And then so we end up with, this is 6.
02:18
100 x over the square root of 4 plus x squared and then minus 500.
02:30
And we want to solve for x for when that is equal to zero.
02:36
So note that the critical values would only be when it's equal to zero because our denominator here is always going to be positive.
02:44
So let's move things over.
02:45
So we get 500 times the square root of.
02:51
4 plus x squared is equal.
02:56
So let's line up my equal sign so it looks a bit nicer.
03:01
Okay, so 500 times the square root of 4 plus x squared is equal to 600x.
03:12
And then now we're going to square both sides.
03:14
So we get 25000000 times 4 plus x squared is equal to 36.
03:24
With four zeros x squared okay so moving things around a little bit we end up with one million is equal to 30 360 000 minus 250 000 so that's 110 000 x squared okay so now that's 1 nil divided by 1100 000 sorry yeah 110 000 i meant and then we want to square root that.
04:06
Okay, so we end up with x is equal to 3 .015.
04:15
Okay, and we again, just double check to make sure that this, in fact, is a minimum.
04:21
So to do that, we do the second derivative test.
04:24
So c double prime of x is equal to.
04:33
Okay, so this is going to be derivative of this thing.
04:40
Okay, so let's do a portion rule...