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Solve the problems in related rates.A plane flying at an altitude of $2.0 \mathrm{mi}$ is at a direct distance $D=\sqrt{4.0+x^{2}}$ from an airport control tower, where $x$ is the horizontal distance to the tower. If the plane's speed is $350 \mathrm{mi} / \mathrm{h}$, how fast is $D$ changing when $x=6.2$ mi? See Fig. 24.24.

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 4

Related Rates

Derivatives

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Harvey Mudd College

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So we have these two planes one going 250 miles per hour towards the airport and another going 300 miles per hour towards the airport where a is going east and bees going north. And we want to find the rate of change of the distance between these two planes when a is 30 miles away and B is 30 b s 40 miles away from the airport. Now what we just have drawn here? Well, it looks like we have a right triangle. So we might want to find some way to shoehorn a triangle property into here. So if I were to call this side of the triangle X this side of the triangle, why and this side of the triangle See, we could use Path a gris to relate all three sides to each other. We have X squared plus y squared is equal to Z script and what we're looking for in terms of X, y and Z is Deasy by D. T. Cause we want to know the rate of change of that red line or the rate of change of seat. Well, if we were to take the derivative of X squared plus y squared is equal disease Weird. We will end up getting dizzy by DT So let's go ahead and do that really quickly So D ready t So all these will look similarly so the derivative of X squared would be to x and then we would have to multiply this by D expert e. T. Because of chain rule were implicit differentiation because X depends on t And then we get the same thing for why so too I times d y bi tt And then this is going to be equal to two z times easy by TT. And now we could divide just by two to get all those accounts alone. All right, so now, before we do anything else, let's figure out what x d exp i d t and why and d y by d. T should be. And we should probably also figure out what Z is as well. So in this case, Z is supposed to be the distance between A and B. So then that means accent. Why would just be the distance between the airport and each of the Plains? So X is going to end up being 30 and then well, the rate of change of this should end up being DX by d. T here. But since the distances decreasing, this should be negative. 250 that we use and then likewise for playing be This will be our why. But then d y by d. T. This here should be negative as well. Since the distance between the plane and the airport is decreasing and will define Z well, we would just need to use Pythagoras again. So go back to X squared. Plus y squared is equal to Z squared and we can plug in that X squared or excess 30. So scoring that would give us 900 and then why was 40 so scoring that would give us 1600. This is even twosies square and well 900 plus 16 hundreds 2500 And then we would want to square root each side so we get Z is equal to plus or minus the square root of 2500. But it doesn't make sense for us to talk about the negative route or Z in this instance, because this is a length between A and B, and it doesn't make sense to talk about a negative distance between two things in this instance. So we have that and then we can actually simplify this to just 50. So we know all of our values for everything outside of Easy buddy t. So let's go ahead and plug everything in. So we said X is 30 DX by DT would be negative 250 And then why is 40 and then d y by DT is negative 300 and we had Z is equal to 50 and then he would have Deasy by d t over here. So the first thing I'm gonna do is just divide everything by 50 because I don't help simplify the numbers a little bit. So doing that would give what we have 30 times Negative five waas 40 times Negative six and this would be equal to easy by DT. I don't want the equal sign here and now we can go ahead and multiply those together so 30 times negative five would be 1 50 and then we add that to 40 times negative six, which should be negative to 40 and adding those together we would get negative 3 90 so minus 3 90 this should be in miles per hour since our other two units for velocity, or, I guess, in this instance speed or in miles per hour. So the distance between these two planes is decreasing at a rate of 390 miles per hour.

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