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2000A television camera at ground level is 2000 feet away from the launching pad of space rocket that is set to take off vertically; as seen in the following figure...

Question

2000A television camera at ground level is 2000 feet away from the launching pad of space rocket that is set to take off vertically; as seen in the following figure_ The angle of elevation of the camera can be found by 9 = tan where € is the height of the rocket: 2000Find the rate of change of the angle of elevation after launch when the camera and the rocket are 5301 feet apart_radians per foot_ Round to five decimal places_

2000 A television camera at ground level is 2000 feet away from the launching pad of space rocket that is set to take off vertically; as seen in the following figure_ The angle of elevation of the camera can be found by 9 = tan where € is the height of the rocket: 2000 Find the rate of change of the angle of elevation after launch when the camera and the rocket are 5301 feet apart_ radians per foot_ Round to five decimal places_



Answers

A television camera at ground level is 2000 feet away from the launching pad of a space rocket that is set to take off vertically, as seen in the following figure. The angle of elevation of the camera can be found by $\theta=\tan ^{-1}\left(\frac{x}{2000}\right),$ where $x$ is the height of the rocket. Find the rate of change of the angle of elevation after launch when the camera and the rocket are 5000 feet apart.

So for these related rates problems it's always a good idea to draw yourself a picture. So I have a rocket and don't laugh at my horrible drawing. But there's my rocket ship And 400 ft away is my tv camera. And obviously that's not a tv camera, that's an old fashioned television. So I've got my rocket ship and the elevation of the camera will change. So we're going to call that feta. And I'm gonna connect this because I'm going to end up with a right triangle. I call this X. Y. And Z. It should be a right triangle where X squared plus Y squared equals the square. So that will help us with the first part of the question. Now, the distance from the camera Is changing at a speed of 600 ft/s. So that's D X. D. T. That's what how the rocket speed is changing and that's 600 feet for a second when it's risen 300 ft or 3000 ft. So when X is 3000, it's changing at this rate and this is going to be a constant. And then I'm asked, how far is the distance from the television to the camera. So I'm asked to find Z. DZ DT, what does DZ DT equal under these conditions? All right. So at that moment It's not 400, sorry, it's 4000 ft left zero. So at that moment X squared three 1000 squared plus Y squared equals E squared. So at that moment for that instant Z will be fixed. So you need to put in your calculator 3000 squared Or recognize that you have a 345 triangle. You recognize that you have a 345 triangle. And you'll be able to recognize that Z will be 5000 ft at that moment. 345. Okay. So now that I have that information, I have to find disease E D T. Which means I'm gonna have to take the derivatives. So going to another screen X squared plus 4000 because remember that's a constant equals Z squared. So taking the derivative, bring down the 22 x times the derivative effects which should be dX DT zero because the derivative of constant is zero to Z. DZ DT I just need to plug in the values that I'm given. So X 3000 dX DT is 600 And Z was 5000. And then to solve for DZ D T or if you wanted to, you could rewrite this formula and divide both sides by two Z. And he would end up with X divided by z times DX GT equals DZ DT. I'll pause. So whether you solve for DZ DT and then substitute in or substitute and then solve DZ D T. Well equal. I'm going to use this up here 1000 Divided by 5000 times 600. Yeah. And we get 360 feet per second. So at that moment, this distance is changing by a rate Of, what did I say? 360 ft her second. And then you're asked how fast is the distance? No. Oh you're asked to find d theta DT how fast is the camera angle changing? Okay. So I still have X, Y and Z. But I'm not going to use easy anymore. And I know that this is 400 or 4000. And this is data. So what's the relationship between these legs of the triangle and data? That would be tangent? So the tangent of theta is X over y. Now 400 is going to be 4000 will be a constant. So I'm going to multiply that by both sides and this is going to be the formula that I am going to take the derivative of. So when I take the derivative, this is a constant. So 4000 times the derivative of tangent seeking squared times the derivative of theta with respect to t. Now I go back to the other page and remember this was 600. This is what I'm trying to find. So I do need one more piece of information here. I need to know data. So I'm gonna come back to this formula and remember X at that moment is three 1000 pete. So I'm gonna solve this for tangent or for to so tha tha will be the inverse tangent of three Force. And I don't know if you're supposed to have it in radiance or in degrees. So because I'm going to leave mine and radiant. So when I do Inverse Tangent of three Force I get approximately 0.64 35 radiance. Okay. Now that I know that I'm gonna plug that in for third up so seek it Squared of .6435. Oops. I didn't leave myself room D. TheTA DT. Which means D theTA DT Will be 600 Divided by 4000. Seek it squared .643 five and plugging that into my calculator. I get approximately .096 Radiance. Uh huh. Second that it's changing.

Hello, Numerator. Welcome back. Okay, so we're in chapter three, Section nine. Page 2 50 number 39. And so you're a cameraman. Yeah. Yeah. And your camera? I can't get away with this thing. So your camera is your 4000 ft from, say, a space shuttle taking All right, So this All right, So and then it's taking off over here, presumably at right angles from the ground from at least initially. Oh, well, I really hate this graphic editor. Yeah. All right. Yeah. Sorry, guys. This is really dominance, huh? Mhm. Let's pretend Okay, already. So So let's say this is where the camera starts out. There's the takeoff point, and this is where the space shuttle ends up. Okay, so this length here, I'm gonna call it X is a constant 4000 ft. You want to know at the moment that why is 3000 ft and apparently the greatest change of of whitey whitey T for the space shuttle is 600 ft per second, At least at this instant at the instant went wise. 3000. You want to figure out how fast this is changing? Pretended to straight line McGuinness. You don't know how fast called the Z Z is changing. Okay, already. So since it's the right triangle, presumably we have I use this movie. Okay, we have Z squared is 4000 square plus y squared because the X because X is not changing. But why is changing? So if you wanna know how fast disease changing find is e d T. So French it implicitly and you get to Z dizzied ET is equal toe now the river of 4000 square toe. It's a constant zero to get to y do anything. So to figure out these DDT Alright, related rates thes e d. T is based on TV while eating, uh, you need to know why which you have. We need to know why Prime which we have. We need to nosy. So what is the well at the moment there, Why is 3000 Z? He is the square root of 4000 squared plus 3000 square. Okay, uh, what's that? 16 million and nine million is 25 million square Bruce of 25 million. Because 1000 squared is 2000, tens of thousands of million and three squared is 9 to 9 million. 10 things. 60 million family. So that reduces to 5. 5000 doesn't. Oh, yeah, it's a 345 triangle. Done. So these 5000 ft. Okay, so I think we can do part A This part. A. Okay, Yeah. 39. Okay. What is wrong with this thing? Three. M'membe has to do with this angle here. Call it feta. A fest that fate is changing. If you're keeping the camera trained on the space shuttle, that is increasing, right? Defended. He would be positive. But right now we're looking for a disease ET so perfect. We're going to say the twos canceled. Right? So now we have 5000 thes e d t or Z. Private life equals 3000 divided T and D Y T t was given a 600. So divide that by 5000 and you get that easy to use. Your crime is 3/5 3000, 5600. And the fifth of 620 times three is 360 pay per second. Yeah. Yes, the exes and wise. We're in feet and the white t was in feet per second. So this is gonna be 360 people. Second Alright, Support be what we have to do with theater So you could talk about the sign of Fada. Is Thea opposite of the hypotenuse 3000 of Zig eso 3000? It's not a constant. So why over Z or you could talk about the co sign? Being 4000 over Z Z is changing. The 4000 isn't changing. We've been talking about the tangent is why over 4000? Because Y is changing with the X is not changing on differentiate That's any one of those expressions. Skip the fate of the tea or think of prime um tangents. Probably nice. Why not? What's tangent? Okay, let's strike tangents. So the tangent of Fada is Why were 4000 so go to one So 39 b The tangent of Fada Mhm his wife over 8000 is not changing. X is not changing, But why is changing all right eso we differentiate both sides implicitly. We expected to using change role we got C can't squared data through attention seeking squared eso through the tangent of stuff is sick and scared of stuff the stuff through stuff change going and this is just gonna be 1 4/1000 of the white GTO and we have the White ET It's still 600 but we don't have is a seeking Well, think about it if she can is one over the Coast Square Seeking is one of the coast and so she can't squares one over the coastline square. Alright, so we divide both sides by seeking square, really multiplying, whether simple we just close side squared So defeated ET is gonna be the coast sine squared of theta over 4000, uh, times they wanted to now do nt t is apparently 600 ft per second and the co sign Well, let's go back to the original diagram Person is the adjacent over the high patterns. So Cartola so that's 4000 at this moment, over 5000 or four fits. Okay, so that's gonna be 1 4000 of 4/5 squared time 600. So that's 6/40 is 3/20 times 16/25. Mhm is is 48 over. Where does that reduce anymore? It does. It produces 2 8/10 4/5, as you just a little bit reduced by four. You got a four appear. You've got a five, man. Okay, so it's gonna be three times four is 12 in the numerator over 505 125. I don't think that reduces anymore. Uh, and that's defeated the Tisa radiance per second, which is 0.96 radiance for second. Exactly. That rounded off. Just run that by my little graphing calculator there. So the theater DT potato prime is exactly 0096 ratings per second. All right, I'm right. That's it for question 39. Hope that was helpful. Good luck with your homework. See you next time. Bye bye.

If we sketch a triangle, we have the camera here in blue and the rocket here in red. And we have the distance between them the vertical height of the rocket and the horizontal distance from the launch pad. So we know this distance is 4000. We'll call this height H and we'll call this distance X. So we have h squared plus 4000 squared equals X squared by the Pythagorean dear. And we're given that the height is 3000 and so that means that X is going to be equal to 5000. Now, if we take the derivative of this equation here, we get to age times th DT He goes to x times dx DT. And if we plug in the values we know we have to times 3000 time 600 equals two times, 5000 times DX DT. And if we divide both sides by two thes cancel and we're left with DX. DT is equal to 3000 time 600 over 5000 and that gives us 1800 over five or 360 feet per second. So 360 feet per second is the rate at which the distance between the camera and the rocket is changing. Now, if we want the rate at which this angle theta is changing, we can take tangent data, which is a church over 4000 Have we take the derivative of both sides? We get seeking square data times di fada t t because th DT over 4000. Now, if we plug in what we know, we have seeking data which is going to be 5000 over 4000 squared times d fader I could see because 600 over 4000. So we have decided DT equals 6/40 times 16/25 which will give us 12 over 125 or 0.96 radiance per second for the rate at which the angle between the camera and the ground is changing.

To the adjacent cited by high parties in the right triangle given in the figure we have coastline of Veda is equal to 800 divided by s. If we differentiate, both sides would respect t on, then get detailed E t on its own and used a changeable We have defense d t is equal to 800 divided by s squared times The sign off t sign of data times D s. D. T. The meteorite detain Aditi in terms of S and P s tiki. Therefore, we need to replace scientific data with an expression in terms of s So since we know that CO sign a theta is equal to 800 divided by s, we also know that sign of data is equal to the square root of one minus co sign squared of failure. So therefore we can rewrite this function as the sign of data is equal to the square root of s square minus 800 squared, divided by s So therefore, defeat Aditi is equal to 800 divided by s square times. Sorry if we simplify this, we get the square root of s square times 800 squared multiplied by s well supplied by D S. D. T


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