00:01
They say that two cars start moving from the same point.
00:02
One travels south at 48 miles per hour, and the other travels west at 20 miles per hour.
00:08
At what rate in miles per hour is the distance between the cars increasing two hours later? and so two cars are starting from the same point.
00:18
One is traveling south at 48 miles per hour.
00:23
So this distance here, which i will call x, is increasing at the rate of 48 miles per hour.
00:30
In other words, dx dt is equal to 48 miles per hour.
00:37
The other car is traveling west at 20 miles per hour.
00:42
So this distance here, i'll call it y, is changing, and the rate of change of y with respect to time, in other words, dy dt is equal to 20 miles per hour.
00:56
And i want to know at what rate in miles per hour is the distance between the two cars, this distance here, i'll call that distance z.
01:08
How fast is z changing two hours later? so my goal is i want to find dz dt.
01:17
So i need to relate the derivatives of x, y, and z.
01:21
To do that, i'm first going to relate x, y, and z without their derivatives, and i'll do so by the pythagorean theorem.
01:27
I can say that x squared plus y squared is equal to z squared.
01:33
And now i'm going to differentiate both sides with respect to t.
01:38
When i do, i get 2x times dx dt plus 2y times dy dt is equal to 2z times dz dt.
01:53
Now after two hours the value of x is going to be 48 times 2 which is 96.
02:00
The value of y will be 20 times 2 which is 40...