00:01
All right, we have two cars going in different directions perpendicular to each other.
00:06
We have car b going north, and he's driving at 60 miles per hour.
00:12
All right, and then we have car a, which was headed west, and he was going 50 miles per hour.
00:17
All right, so i have the basic setup here for us, and it says they're both driving towards the intersection of the two roads.
00:26
So our two roads, a's road goes this way, b's road goes this way, and they are at different points on those roads.
00:34
And we want to figure out at what rate are they approaching each other.
00:38
We want to know at what rate is the distance between them changing.
00:43
That's what we're looking for here.
00:45
All right.
00:46
So i already went ahead and labeled a little bit of our information.
00:49
They told us that we're looking for how fast that distance is changing when b, this distance from the car b to the intersection, is 0 .4 miles, 4 tenths of a mile.
00:59
And the change in b is 60 miles per hour.
01:01
Now, i'm going to go ahead and put in that that is a negative change.
01:05
He's driving at 60 miles per hour.
01:07
That's his speed.
01:08
But b is getting smaller.
01:10
This distance is getting smaller.
01:11
We would consider that a negative change.
01:13
So that's going to be a negative 60 miles per hour.
01:17
Same thing with a.
01:19
A, they wanted it to be 3 tenths of a mile from the intersection.
01:23
And it's going towards the intersection, which means that that distance, of a is getting smaller, so i'll count this as a negative 50 miles per hour.
01:32
Notice the notation, that's the change in a with respect to time because it's per hour.
01:37
So we're talking about the change in that length with respect to time.
01:42
All right.
01:42
So when we finish drawing that, we put our roads in.
01:45
Basically, what we're left with is a right triangle.
01:49
And what often works well when we're trying to find a formula deal with a right triangle? pythagorean theorem.
01:54
And that's what's going to work for us here as well.
01:57
I'm going to call this length a because that's where car a is, and we know how long that one is.
02:03
So i'm just going to go ahead and write a squared plus, and i'll call this guy b squared.
02:07
So a squared plus b squared is equal to, and that's going to allow me to solve for this length, which is c squared.
02:16
Now, i'm not really looking for this length, but i'm going to need it in the end.
02:21
What are we looking for? we're looking for how fast the distance between them is.
02:25
Changing so i'm not really looking for c that's not the answer to the question i'm looking for how fast is c changing with respect to time so before i go and calculate what the value of c is let's see if we can figure out what what a formula is for dcdt well we have a nice formula to use here so what we're going to do is we're going to take the derivative of this thing so we're going to differentiate this with respect to time so i want to know what is the change in this thing over time all right, so we take the derivative of a squared.
02:58
Now, a is a variable.
02:59
It is certainly changing, so i can't think of it as a constant at 3 tenths of a mile because it is changing.
03:05
So when i take its derivative, it's 2a.
03:09
And then because of the chain rule, we're taking the derivative with respect to time.
03:13
So we have times the change in a over time...