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
Success IS NOT What you Think it Will Be
So you do not rule out goals because you think they are unattainable? That’s one of your principles? Yeah, so let me clarify that. Until you’re on the journey, you don’t know enough about it. So when you try to assess, can I be successful or not be succe…
Energy graphs for simple harmonic motion | Simple harmonic motion | AP Physics 1 | Khan Academy
What I have drawn here is a mass sitting on a frictionless surface that is attached to a spring that is attached to the wall. What we’re going to do is we’re going to compress the spring; we’re going to get the mass to position A. Right now it’s at positi…
Presenting: Greeking Out by National Geographic Kids | Podcast | Overheard at National Geographic
Foreign last week, you heard our episode on King Tut. To help us keep the ancient Egyptian party going, we’re welcoming the Greeking Out podcast from Nachio Kids. They have a special episode dedicated to another Egyptian pharaoh and mythmaker. Here to hel…
The Origin of El Chapo | Narco Wars
[music playing] It was find everybody involved. Find them now. We knew there was an individual that was responsible for all the logistical movement of marijuana and then cocaine, but we weren’t sure who he was. So we raided house, after house, after hous…
❄️🇬🇧 London Snow Day 🇬🇧❄️
Wow, it finally snowed again in London! A snow day not to be squandered inside. I’m supposed to be working today, but does daily vlogging count? I’m not a daily vlogger, but I think if I make a vlog, that can totally count. Come join me as I do nothing m…
Solving equations by graphing: intro | Algebra 2 | Khan Academy
We’re told this is the graph of y is equal to three halves to the x, and that’s it right over there. Use the graph to find an approximate solution to three halves to the x is equal to five. So pause this video and try to do this on your own before we work…