yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

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


3m read
·Nov 11, 2024

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.

More Articles

View All
Marcus Aurelius - Stop Caring What People Think
In Meditations, Marcus Aurelius said, “Don’t waste the rest of your time here worrying about other people—unless it affects the common good. It will keep you from doing anything useful. You’ll be too preoccupied with what so-and-so is doing, and why, and …
Optimistic in India | Years of Living Dangerously
I was told that if I wanted to see how the US will play a part in India’s energy future, I should come here—a coal power plant, believe it or not, erected right in the middle of Delhi. Mr. Ambassador, nice to meet you. Nice to meet you. I love the movie …
40 years of experience with corporate jets.
I’ve noticed that there is a few other aircraft around. This isn’t it? Yeah, it’s busy ramp right here. This is a Gulf Stream 450 over here. Wait, wait, wait! Steve, how do you know that it’s a Gulf Stream 450? After 40 years looking at these things, I k…
Gideon v Wainwright
[Instructor] Now we’re gonna talk about an important Supreme Court case that reinforced the idea that states have to provide the same rights when people are arrested and accused of a crime, as are guaranteed in the United States Constitution. And that c…
Finding the Titanic | Bob Ballard: An Explorer’s Life
The Titanic was really a cover for a highly classified military operation to investigate two nuclear submarines that we lost during the Cold War with all hands: the USS Thresher and the USS Scorpio. They ironically turned out to be on either side of where…
Graphs of MC, AVC and ATC
In the previous video, we began our study of ABC Watch Factory, and we tried to understand the economics of the business based on some data that we had already collected on our costs and how much output we can produce based on how many labor units we had.…