Analyzing related rates problems: equations (Pythagoras) | AP Calculus AB | Khan Academy

Two cars are driving towards an intersection from perpendicular directions. The first car's velocity is 50 kilometers per hour, and the second car's velocity is 90 kilometers per hour. At a certain instant ( t_0 ), the first car is a distance ( X_{t_0} ) of half a kilometer from the intersection, and the second car is a distance ( Y_{t_0} ) of 1.2 kilometers from the intersection. What is the rate of change of the distance ( D(t) ) between the cars at that instant?

So at ( t_0 ), which equation should be used to solve the problem? They give us a choice of four equations right over here. So you could pause the video and try to work through it on your own, but I'm about to do it as well. So let's just draw what's going on; that's always a healthy thing to do.

Two cars are driving towards an intersection from perpendicular directions. So let's say that this is one car right over here, and it is moving in the direct x direction towards that intersection, which is right over there. And then you have another car that is moving in the y direction. So let's say it's moving like this.

So this is the other car. I should have maybe done a top view. Well, here we go. This square represents the car, and it is moving in that direction. Now they say at a certain instant ( t_0 ), so let's draw that instant. The first car is a distance ( X_{t_0} ) of 0.5 kilometers, so this distance right over here, let's just call this ( X(t) ), and let's call this distance right over here ( Y(t) ).

Now, how does the distance between the cars relate to ( X(t) ) and ( Y(t) )? Well, we could just use the distance formula, which is essentially just the Pythagorean theorem, to say, well, the distance between the cars would be the hypotenuse of this right triangle. Remember, they're traveling from perpendicular directions, so that's a right triangle there.

So this distance right over here would be ( X(t)^2 + Y(t)^2 ) and the square root of that. And that's just the Pythagorean theorem right over here. This would be ( D(t) ), or we could say that ( D(t)^2 ) is equal to ( X(t)^2 + Y(t)^2 ).

So that's the relationship between ( D(t) ), ( X(t) ), and ( Y(t) ), and it's useful for solving this problem because now we could take the derivative of both sides of this equation with respect to ( t ). We’d be using various derivative rules, including the chain rule, in order to do it. This would give us a relationship between the rate of change of ( D(t) ), which would be ( D'(t) ), and the rate of change of ( X(t) ), ( Y(t) ), and ( X(t) ), and ( Y(t) ) themselves.

So if we look at these choices right over here, we indeed see that ( D ) sets up that exact same relationship that we just did ourselves. It shows that the distance squared between the cars is equal to that ( x ) distance from the intersection squared plus the ( y ) distance from the intersection squared. Then we can take the derivative of both sides to actually figure out this related rates question.