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Related rates: Approaching cars

As two cars approach the same intersection from different roads, how does the rate of change of the distance between them change? Created by Sal Khan.
Video transcript
So this car right over here is approaching an intersection at 60 miles per hour. And right now, right at this moment, it is 0.8 miles from the intersection. Now we have this truck over here, it's approaching the same intersection on a street that is perpendicular to the street that the car is on. And right now it is 0.6 miles from the intersection. And it is approaching the intersection at 30 miles per hour. Now my question to you is, what is the rate at which the distance between the car and the truck is changing? Well to think about that, let's first just think about what we're asking. So we're asking about the distance between the car and the truck. So right at this moment, when the car is 0.8 miles from the intersection, the truck is 0.6 miles from the intersection. The truck is traveling at 30 miles per hour towards intersection, the car is travelling at 60 miles per hour towards the intersection right at this moment. What is the rate at which this distance right over here is changing? And just so that we have some variables in place, let's call this distance s. So what we really are trying to figure out is right at this moment, what is ds dt going to be equal to? Let's think about what we know that we could use to somehow come to terms or figure out what ds dt is. Well we know the distance of the car and the intersection. And let's just call that distance, let's call that-- I don't know, let's call that distance y. So y is equal to 0.8 miles. We also know that-- so let me write this-- we know that y is 0.8 miles right now. We also know the dy dt, the rate at which y is changing with respect to time is what? Well y is decreasing by 60 miles per hour. So let me write it as negative 60 miles per hour. Now similarly, let's say that this distance right over here is x. x is 0.6 miles right at this moment. So we know that x is equal to 0.6 miles. What is the rate at which x is changing with respect to time? Well, we know it's 30 miles per hour is how fast we're approaching the intersection, but x is decreasing by 30 miles every hour. So we should say it's negative 30 miles per hour. So we know what y is. We know what x is. We know how fast y is changing, how fast x is changing with respect to time. So what we could try to do here is come up with a relationship between x, y, and s. And then differentiate that relationship with respect to time. And it seems like we have pretty much everything we need to solve for this. So what's a relationship between x, y, and s? Well we know that this is a right triangle. The streets are perpendicular to each other. So we can use the Pythagorean theorem. We know that x squared plus y squared is going to be equal to s squared. And then we can take the derivative of both sides of this with respect to time to get a relationship between all the things that we care about. So what's the derivative of x squared with respect to time? Well, so you're going to need the derivative of x squared with respect to x, which is just 2x, times the derivative of x with respect to time, times dx dt. Once again, just the chain rule. Derivative of something squared with respect to the something, times the derivative of the something with respect to time. And we use similar logic right over here when we want to take the derivative of y squared with respect to time. Derivative of y squared with respect to y, times the derivative of y with respect to time. Now on the right-hand side of this equation, we once again take the derivative with respect to time. So it's the derivative of s squared with respect to s, which is just 2s. Times the derivative of s with respect to time. Once again, this is all just an application of the chain rule. So now it looks like we know what x is, we know what dx dt is, we know what y is, we know what dy dt is. All we need to figure out is what s and then what ds dt is, the rate at which this distance is changing with respect to time. Well what's s right now? Well we can actually use the Pythagorean theorem at this exact moment. We know that x squared-- so x is 0.6-- we know that 0.6 squared plus y squared, 0.8 squared is equal to s squared. Well this is 0.36, plus 0.64 is equal to s squared. This is 1, it is equal to s squared. And we only care about positive distances, so we have s is equal to 1 right now. So we also know what s is. So let's substitute all of these numbers in and then try to solve for what we came here to do. Solve for ds dt. So the rate at which, so 2 times x-- maybe I'll do that in yellow-- 2 times x, x is 0.6 so it's going to be 1.2 times dx dt, so that's negative 30 miles per hour. So times negative 30 miles per hour, plus 2 times y is 1.6, times dy dt is negative 60 miles per hour. And I'm not writing the units here. But if you were to write the units, you will see that all of our distances are in miles. And all of our time is within hours, so we're going to get an answer when we solve for ds dt that's miles per hour. But I encourage you, if you want to, to actually write out the units and see how they work out. And so this is going to be equal to 2 times s. Well, s is 1 mile, so it's just going to be 2 times ds dt, which is what we're trying to solve for. So what do we get here on the left-hand side? So 1.2 times negative 30, that's negative 36. Right? 1/5 of 30 is 6, yep that's right. And then 1.6 times negative sixty, that's going to be negative 96, is equal to 2 times ds dt, is equal to 2 times the rate at which our distance is changing with respect to time. On the left-hand side right over here, this is negative 132. Negative 132 is equal to 2 times ds dt. Divide both sides by 2, we get negative 66. And now we could put our units if we want, miles per hour is the rate at which our distance is changing with respect to time. So ds dt is negative 66 miles per hour. Does it make sense that we got a negative number here? Well sure, this distance is decreasing right at this moment, as they approach the intersection.