00:01
In this question, we are told that oranthologists determine that birds don't really like to fly over water because it takes more energy to fly over water than over land.
00:12
We have a situation in which birds are on an island that are at least five kilometers from a point in land that will call b, and it will fly to some point c and then fly to its nest line at point d.
00:27
And we know that the points b and d are 13 kilometers apart.
00:31
So we're told that it takes 1 .4 times the energy to fly as much over the water and as of land, and we want to minimize the amount of energy expended for this.
00:44
So in order to do this question, let's consider this distance here, which we'll call x.
00:50
The distance bc is x.
00:56
So therefore, we can say also cd is 13 minus x.
01:02
We can also come up with a situation for the amount of distance from the island to point c, whatever it is.
01:11
By the pythagran theorem, it's equal to 5 squared plus x squared, or just a square root of 25 plus x squared.
01:23
So we know that the distance to travel, the distance i travel is going to be this distance drawn over in blue.
01:39
So i want to go like this as such.
01:42
And let me try to travel.
01:43
Draw direction of motion arrows for you.
01:48
So i know that the distance we have to travel is equal to square root of 25 plus x squared, added to 13 minus x.
02:01
And we know that it's going to take 1 .4 times the amount of energy it takes to fly over land.
02:07
So we're going to call that constant k.
02:12
This determines e of x, the amount of energy that's going to be taken is going to be 1 .4k, being multiplied, by the square root of x squared plus 25 plus 13 minus x.
02:41
We'll distribute the ks itself.
02:48
Flying over the water, it actually costs 1 .4k.
02:52
Flying over the land only costs k energy.
02:55
So the energy won't be written like that.
02:58
It'll have to be written like this.
03:15
Taking the derivative of this, because we'll have to do so anyway, we get that e prime of x will be equal to 1 .4.
03:28
4k times x divided by x squared plus 25 minus k.
03:46
And the first derivative is just remember that we have to use a chain rule here.
03:51
And when we combine the fractions, we're going to get that this is going to be equal to k times 1 .4x minus the square root of x squared plus 25, all divided by the square root of x squared plus 25.
04:09
And we want to set this equal to 0, since the denominator will never be zero, we only really have to consider this equaling 0, 1 .4 times x minus square root of x squared plus 25, and where that equals 0.
04:29
And by solving this equation for x, we're going to have that x will be equal to the positive square root of 25 over 0 .96, which is about 5 .1.
04:47
And we know that this is going to be a minimum.
04:51
So what we want to do is we want to fly 5 .1 kilometers right of the point b.
05:06
Now the second part of this question, let's w and l denote the energy to fly over water and land.
05:15
And we want to figure out what values of w over l mean...