00:01
All right.
00:03
So for this problem, we've got a lot of moving pieces.
00:08
So let's get started.
00:10
First of all, we've got a river.
00:13
It's two kilometers wide as shown over here.
00:16
All right.
00:17
We've got a boat.
00:18
I've labeled it in blue.
00:19
I've put it on this river in blue.
00:21
And it wants to cross the river and hit this red point.
00:25
That's one kilometer upriver.
00:27
Okay.
00:29
But we've also got the river flowing.
00:31
And it flows.
00:32
At 2 kilometers per hour downriver and the boat can only move at 10 kilometers per hour all right so a lot of different things going on here but it's a vector problem so let's first start thinking about vectors okay so first things first let's think about this boat it wants to go a certain direction all right and so we're going to make this vector give it an x -ququoise coordinate and a y coordinate.
01:07
Okay, that's how we're going to describe this vector for now.
01:10
This is where the boat is trying to move.
01:12
Okay, it's also going to be affected by the river.
01:15
All right, but just to visualize this too, and we'll come back to this later, all right.
01:22
We've got our x coordinate, our y coordinate.
01:25
Both of these are describing how far, how fast we're going in the x direction, how fast we're going in the y direction, because altogether we're going 10 kilometers per hour.
01:38
All right.
01:41
And so it's this hypotenuse that describes our overall motion that the boat is trying to make before it's getting slowed down by the river.
01:48
Okay.
01:50
After it's getting slowed down by the river, we're going to have something different.
01:54
Here, i'm going to do this in red because hopefully by the end of this, this will take us straight to the destination.
02:03
Okay.
02:05
But in the red, our y coordinates staying the same.
02:10
Nothing's pushing us back towards one of the banks or anything.
02:14
But we are getting pushed downriver.
02:17
All right.
02:17
And so we're not going to get our full speed here.
02:19
We're going to get two kilometers per hour less than what we're trying to go in the next direction.
02:28
So from here we have like the red vector is really our vector of movement.
02:33
Okay, it is really where we're going, how the boat's moving, whereas this blue one is what the boat's trying to do, okay, where we're pointing the boat.
02:45
So from here, one more thing we need to notice is that now we have a relationship over here from this triangle i've drawn here.
02:58
We have a relationship between x and y, right? because we know this is a bright triangle.
03:04
All right and so we know a squared plus b squared equals c squared so we have that x squared plus y squared is equal to 10 squared or 100 all right um and we're going to go ahead and solve this for why because i'm going to actually plus change y in this uh vector here okay so let's subtract x squared over so y squared is equal to 100 minus x squared all right, and then we'll have to square root.
03:37
So y is equal to the square root of 100 minus x squared.
03:42
Okay.
03:45
So this vector over here now, we can just plug this in.
03:49
X minus 2, square root of 100 minus x squared.
03:57
Okay, that's our vector.
03:59
That's our vector of our actual movement.
04:02
Okay, so let's go back up here and remember that we want to go two kilometers this way, one kilometer this way.
04:12
All right.
04:13
What this means, back down here at our movement vector, is that our x coordinate, which is how much we're going to move to the right, we want it to be half of how much we move up.
04:27
Okay.
04:29
One way to write this is that we want our x coordinate, x minus 2.
04:36
All right.
04:38
We want it to be half of the y4.
04:45
That's just writing my words as an equation.
04:50
From here, we can solve this.
04:53
All right, so we're going to multiply by two on both sides.
04:57
So 2 times x minus 2 is equal to square root of 100 minus x squared.
05:05
All right, we need to distribute that too.
05:08
I made this video before and didn't actually distribute it.
05:12
I had to redo everything.
05:14
So make sure to distribute it.
05:17
Next, we're going to square everything to get rid of this radical.
05:26
Make sure you distribute this part.
05:30
This becomes 4x squared minus 16x plus 16.
05:39
All right, and now the square root sign is gone from the other side.
05:46
Wow, that was amazing too there.
05:52
We're going to move everything to one side.
05:54
So subtract by 100, add x squared.
06:02
Minus 84 to 0.
06:06
All right, so this is a quadratic equation we can solve.
06:09
It can be factored.
06:10
You could also use the quadratic formula.
06:14
All right, but it can be factored into this.
06:19
X minus 6 times 5x plus 14.
06:22
All right, so this gives us two solutions.
06:25
X minus 6 equals 0.
06:27
5x plus 14 equals 0.
06:31
All right, so x equals 6 is one solution.
06:34
That's a good sign, right? we did want to move right, so we want our solution to be positive...