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Aplane flying horizontally at an altitude %f 2.2 miles and at a speed of 555 mi/hr passes directly over rdar station- Find the rate at which the distance from the p...

Question

Aplane flying horizontally at an altitude %f 2.2 miles and at a speed of 555 mi/hr passes directly over rdar station- Find the rate at which the distance from the plane to the station is increasing whenothe distance between the plane and the station is 6.9 miles_ If necessary, round your answer to two decimal placesVariableDescriptionUnitTime Horizontal distance the plane has traveled since time Distance between the plane and the ground at time t Distance between the plane and the radar station

Aplane flying horizontally at an altitude %f 2.2 miles and at a speed of 555 mi/hr passes directly over rdar station- Find the rate at which the distance from the plane to the station is increasing whenothe distance between the plane and the station is 6.9 miles_ If necessary, round your answer to two decimal places Variable Description Unit Time Horizontal distance the plane has traveled since time Distance between the plane and the ground at time t Distance between the plane and the radar station at time t dr Rate at which € changes with respect to time 2 Rate at which y changes with respect to time dz dt Rate at which 2 changes with respect to time mifhr milhr milhr The rate at which the distance is increasing is about



Answers

A plane flying horizontally at an altitude of $ 1 mi $ and a speed of $ 500 mi/h $ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is $ 2 mi $ away from the station.
(a) What quantities are given in the problem?
(b) What is the unknown?
(c) Draw a picture of the situation for any time $ t. $
(d) Write an equation that relates the quantities.
(e) Finish solving the problem.

Alright for this exercise, We're considering the motion of two different planes. So we have a Cessna. I'm going to label see heading south at 120 MPH. And we have a Boeing 7 47 heading west Boeing. I'm labeling with B that's heading wet. Whoops. Wrong direction that should be here heading west at 600 MPH. Yeah, eso They're flying towards the same point at the same altitude. We're also told that our Cessna is a distance of 100 miles from the point, while our mowing is a distance of 550 miles. So obviously our little sketch there is not to scale. So what we're going to do for this problem here is first of all, we want to set up equations that will describe the positions of these two planes. So what I'll do is we'll have C equals zero. I had plus 100 minus 1 20 t j hat. If that notations unfamiliar, it's sort of just like a vector notation or a particular form of vector notation. So, alternatively, I could just write it as pair of coordinates. So we're saying that the Cessna is in line with the point, so it's distance horizontally will always be zero. It starts 100 above, but it's getting closer at a rate of 120 per hour than for B. We are starting 550 miles west. We're sorry we're starting 550 miles east of the intersection Point, so we start out at 5 50 and, because we are heading westwards, will be subtracting from our position. So we'll have 550 minus 600 t and then there's no vertical motion. So that's just going to be zero for Part B. We want to figure out on equation that will give us the distance between the two planes. So we know that we have our Cessna. You have our point. What's called that? Oh, it's called that C. It's called E. The distance between the two planes. It's going to be the high pot news of this right angle triangle that's formed here so the total distance will be equal to I'll put this as brackets with to the power of one half outside, rather than doing this square root over everything. Um, there will be the distance from B to the origin square, plus the distance from sea to the origin squared. MEPs squared all to the power of one half. So in this case, you would have 100 minus 1 20 t squared plus 5, 50 minus 600 t squared all to the power of one half. So one moment as the inside of the square root, expanding everything out and simplifying, we should get 374,400 t squared, minus 684,000 T plus 312,500 still all to the power of one half. Now, for part C, we want to plug this into a graphing utility and figure out what the minimum distance between the planes are. So I'm just going to jump over to a graphing utility. So loading up into days, most app, you can also run this just a Web page. Plug in. We'll have everything under square root. Then we have 374,000 374,400 rather X squared, minus 684,000 x plus, 312,500 which should give us our plot of the distance. Plugging this into the graphing utility is actually part C. And then Part D is answering. When did our how close do they get and when do they get that close? So we can see that the closest that they'll get is at about 0.9135 and they get about 9.806 miles away from each other as last part. We want to simulate this by plugging in the formulas that we derived for part A and plotting them simultaneously. So the way that we can do that using Dez most here is what I'll do. So plug in the coordinate form. So we had for C was zero plus 100 minus, we would have won 20. All right? Actually, I believe Okay, Yeah, so we would write 120 t. But for the sake of being ableto see how the planes are moving relative to each other Instead, I put in 120 a, uh, which makes Dez most put in a slider for a So that means that basically, I'll be able to slide a long time. We'd be able to see the emotion of the two planes. So label that Cessna and I'll also adjust the time range that that a varies over. So we'll be going from zero upto. One should be able to see the starting position for assessment. Yet there we go. And then similarly, for the Boeing you would have it was 5 50 minus 600 a and then zero. The vertical axis enable a label of the Boeing and need to adjust our view Here. There we go. So we have our Boeing in our assessment being plotted at the same time. And now we can watch. Of course, it looks like the Cessna is moving faster, but that's just due to a distortion. Because of the scaling, we see that the Boeing is in fact going much faster, But it will pass by that, um, intersection point of their flight paths a little bit after the settlement passes by. So no air tragedy crisis averted

In problem 29. An air traffic controller. What's two planes complain When the X axis here at 205? Yeah, we're from the origin. The first Blaine. He's at 250 miles. And the second plane is on the Y axis at 300 300 miles. The first blame goes with the speed. Civility equals 400 and 50 MPH. Here, this is 200 25. And this plane goes in this direction with a velocity equals 600 miles. We're all we can see that after just half a now, er they will coincide here because they have the same altitude we're about to eat. We want to calculate the rate off the distance between the brains. This is the distance there is here. A distance? Yes. We want to calculate its decreasing. Great To do so. We have here X and we have here. Why? And we can get the s in terms of X. And what? Yes equals square root off X squared plus y squared. And here we want to get the s by DT. We can differentiate both sides with respect to the tie. It equals the differentiation of a square root is one divided by two multiplied by the square root and we multiply by the differentiation off the blown Amiel under the square root which is differentiation of X squared is two x multiply it, boy the X by DT because we differentiate with respect to the time Plus why square differentiation is toy. But the blue eyed boy do you worry by DT Because we differentiate with respect to time, Then to get the s by d t it X equals 225. And why equals 300 x equals 225 on Doughboy equals 300 miles equals to multiply by 225 Multiply by the velocity the X and you ve boy 450 plus two multiplied by well, you, which is 300 multiplied by 600 MPH, divided by two but apply by square root 225 plus 300 equals before we calculated we have here X squared and y squared. Then the answer is 700 and 50 miles. But, uh oh for Bharti. We want to calculate the time that the air controller as toe get one off the planes in a different path, not to go inside here. You can see that we can calculate the time which equals the distance, divided by the velocity full. Any off the planes, we can take the distance. 300 divided by the velocity. 600 equals half an hour. Then the controller. Yep. Have half an hour to move one of the planes out of its both. And these are the final answers off our problem.

With related treats. It actually some quantity is changing by relating the quantity to other quantities who started off Things are known. So we have this picture or exercise 24. So he been the ex absurdity Sequels to 300 miles per are converting it to feet her second. You need to multiplied by one Albert tasks and second and multiplied by but that a defeat by miles one month. So it is equivalent to 444 seconds. So we have the equation off tangent that is about opposite of a decent cider in the Italian Angeles equals to 4000 of our X. Then we've crossed motive. Play it. We have X equals toe 4000 of Burton Get data or you simply concluded Thesis 4000 co Italian Tetteh solution for later A. We need to determine that data over time at data calls by over six. Or that it's 60. Our turkey hurting significance. Also cost of Palmer six. So we have X close 4000 times gotta get data. So we have the little body off Exxon where they they did that it's over time. So we have because the 4000 times negative because he can skirt at the times detect ability. So substitute the value of the eggs ability that support our foreign party in the valley off the data angle, which is because two power six six. So we have to. So we have 4 4400 body cause negative 4000 thanks to Scott Data with that, that's how we have the data over the T that it's equals toe were important over 4000 times four and we have this one LePen over 404 100 per seconds. All these humility equals two in degrees. So we have negative level for over branded. That's 180 of our pie still become causing neck data that heated up politicals negative 1.576 Now we have this another creation which states that on angle data east. Now with my parable, we have the undersigned data that its Sequels toe opposite over haughtiness. So what does number life represent? Like stuff But in this year for you How what waas that possible to play it So I called sport in the world science data. So we have no I called support thousands Kasey Kahne data. So we have the Web elitist, unknown and that 30 degrees or by oversexed region. So they have a difficult negative 11 over 100. So we have signed number 61 cause one half cycles. And over 66 credited Roberto and content then five or six that this goes to Scott of the three. So they were abilities equals 4000 times negative doesn't get at times, got at the data the data D. So you have subjected the gave bank, so 4000 times negative cost you can buy over six times pathetic number six temps. Negative. 11 over 400. So we have 4000 times two times recorded toe three times negative. 11 over 400. So the wire. But it is because transit 81 considered five. So the answer is this one in the sun for a and the D. C. Four b

And this problem, we're told that our air traffic control of spots to airplanes at the same altitude, converging to a point as they flat as they fly it, right angles to each other. One airplane is 150 miles from the point and has the speed of uh 415 MPH. The others 200 miles from the point and has the speed of 600 MPH. Yeah, we're taught to calculate the rate at what rate is the distance between the plane changing and how much time does the controller how to get one of the planes on a different flight Pat. So the first thing to do is to write out what you do know and the problem and we know that let let the let the distance of one plane from the point of convergence B characterises X miles X mouse. And that that this is that the second plane from the point of conversions be characterized as y mouse and the distance of the two planes b characteristics Z mouse. So with that being said we know that X. Will be equal to 150 miles. And we know that this speed which is D X D. T would be equal to negative 150 MPH. So we know speed which is dX DT will be equal to negative 450 miles. Oh do you know why? For the for the second for the second plane you know why is giving us 200 miles? Mhm. Yeah. And we know that the speed also the same as DX DT. Dy DT because the axis change Dy DT would be a hunt 600 MPH negative 600 miles. Yeah. Yeah. So that's what we do know from the problem. Now we know that the pythagorean um is X squared purse why squared is equal to Z squared right now? We could actually, and let's say this is equation line. Okay, Actually, differential take the differential of this equation. If we do that we have to X plus two. Why is equal to to the so were taken two X dx DT plus to I Dy DT is equal to to Z DZ DT. So we're taking that with respect to Z. Y N X. That's why we need to have that. Yeah. Mhm. And you could say this could be your another equation. Right? So now the reason we did this, so now we could easily input the values for X. And the values for why? So sub teach Children ex N. Y. We have two X. Two X becomes well, yeah, let me put that into my calculator first. Let's take this equation. X becomes X. Is 1500. Mhm squared flowers, 200 squared because equals two Z squared. Okay, right, so now we can actually find a Z. So 1500 square it becomes 22,500. Yeah. Okay. And 200 squared gives you 4 40,000 equals Z squared. Yeah. We left towards 60 62,500 first. What Z equals Z squared. Now we could take the square root of both sides to isolate Z. And we have Z becomes, yeah, the square root of 6500. The square root of 62,500. Mhm. Um Okay. Mhm. That becomes 2 50. Mhm. And we know that is 250 miles. So now we know our X, Y and Z values. Now we could easily, yeah. Um substitute X. DX DT Why do you I D T. And DZ DT. The only thing we don't know at this point is DZ DT. But we know D. X. D. T. And D. O I D. T. So we subject to answer these equations, equation we have to we know X. Is 1 50. Okay. We know our D. X. D. T. Is negative 4 50. We know why it is 200 and we know Dy DT is negative. 600 is equal to. We know Zs 2 50 and we don't know what R. D. X. D. T. Is DZ DT but in that and see a calculator. Okay. We left word negative 13,000 135 1000 minus 240,000 equal to uh 500 D. C. D. T. All right. Yeah. We got a negative mm to isolate DZ DT. We divide both sides by 500. We have these E. D. T. Becomes yeah negative 750 negative 715 MPH. So this tells us that the distance between the two planes changes speed constantly at negative 140. They give 715 MPH. That's your part A that's the rate of change. So for part B. Uh huh. For B we're told to calculate how much time does the controller have to get one of the airplanes on a different flight pack. Okay. We know that uh time taking where the planes converge at a reach at a certain point of convergence. Is this an over speed? So time taking right time taken for the Plains? Uh huh. For both planes to reach the point of convergence, distance over speak. Oh. And what do we know? We know that our distance 150 miles and we know that our speed, it's 450. Okay? And time is neither positive or negative. Time is always positive. So these values are going to be positive. Okay? So for for like the speed ignore the negative values because time is always positive. So you have well 50. Yeah. Oh over 4 50 which is 1/3 hours. Uh huh. It was it's weird seeing one of the three hours. Uh huh. So you can actually convert this two minutes. We know there are 60 minutes in one hour. So you could say 1/3 times 60 you have the three cancels into 60 you got 20. So you look for 20 minutes. Yeah so this tells us that the time needed for the planes are all converged to a certain point, they would take 20 minutes. It so there you have it


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