m-to-measure-the-takeoff-performance-of-an-airplane-the-horizontal-position-of-the-plane-was-measure

Question

[M] To measure the takeoff performance of an airplane, the horizontal position of the plane was measured every second, from $t=0$ to $t=12 .$ The positions (in feet) were: $0,8.8,$ $29.9,62.0,104.7,159.1,222.0,294.5,380.4,471.1,571.7$ $686.8,$ and $809.2 .$
a. Find the least-squares cubic curve $y=\beta_{0}+\beta_{1} t+$ $\beta_{2} t^{2}+\beta_{3} t^{3}$ for these data.
b. Use the result of part (a) to estimate the velocity of the plane when $t=4.5$ seconds.

Pawan Yadav · Accepted Answer

Okay, so question is horizontal position measured every second from 0 to 12. 2nd is given as this in feet. Okay, so we have total 12 data or 13 Data&#x27;s given in feet. Were you to finally square cubic off model by this equation? And also the velocity at time T equals to 4.5. So looking at this situation, we can model. This is why capital wiry President Vector y is here. We&#x27;ll have tea vector. Okay, let me write. This is why this is t metrics. And then we have vector off unknown parameter that is Betta. So this is the form that we have now we can interest in this position to write t metrics beat a vector is equals toe. Why vector right? So we can put the value and have our so t metrics is representing what won t t square t Q in rose. Okay. And number of data here and then we have bitta, which is representing better not bitter. One bitter to and vita three. And here we have. Why one y two y three. Okay, Okay. So now we can put the data and have our metrics. 40 equals to zero we&#x27;ll have 1000 40 equals toe one, We&#x27;ll have 111 square is 11 Q B is 1 40 equals toe to we&#x27;ll have your one than to to square, that is 42 cube eight, then 13 and three Square 93 cube 27. Then we have four. So it will be 144 square, then for Q 64 it will go till the last value. That is 12 seconds. So 1, 12, 12 square and then 12 cube 17 to it. Okay, And here we&#x27;ll write this as bitter only. And here we&#x27;ll have our why value? Why is the output back? That is zero 8.8 29.9 62.104 point seven and it will go till the last value 809.2. Okay, So to solve these metrics, we have to multiply with transpose off this. So if this is our why Betta and let Sorry this was a t right? So t be tha and this was our y. So we have to multiply transpose off tee times t b tha and this will be transposed. Why, Right? So transpose off TV are multiplying toe Make it square metrics. So on multiplying this we get so in calculating this dy transposed t we get 13, 78 78 Okay. And 6, 56 50 in 60846084 Then we have here 6. 50. Then it is 6084 6084 Then we have 60 710 And here we have six Zito 71060710 Then we got 630708 Okay, again here. 630708 And the last value 6735950 times beater is equals to metrics. We got multiplying. This or vector, we got multiplying. This is equals to 3800.2351 to 7.7348063 point nine and 3599800.9. Okay, so now we need beater so we can take in words in words off this metrics both side. So if we assume this metrics is let&#x27;s say a okay. Times beater. And let&#x27;s suppose this is, uh see. So we need to multiply by a In was each side so that we are left with I which is nothing but one in case of metrics. And here we have a first time see so on, calculating inwards and multiplying With this we get Vita is equals to do it minus 0.85 4.7, 5.5 five and minus 0.2 Okay, so that is a pita. Just nothing. But so now we can write our equation y s beat or not, which is 0.85 plus beat a one So 4.70 plus bitter to which is 5.55 t square plus bitter three, which is minus 0.2 de que So this is an answer for part A Not to calculate velocity for part B, we should differentiate this with respect to time. So why dash is equal to differentiation of this is 4.7 times things different Station is one, so we&#x27;ll have to 4.7. This is 5.55 times to T and minus 0.2 times three t square. So this is why dash now we need to find white ash. At T equals to 4.5. So why&#x27;d I saw 4.5 is equals to 4.7 plus 5.5 into two is let us carefully directly off the values ts 4.5. Okay. Here. Minus 0.2 into three in 24.5 square. So we get why that&#x27;s 4.5 is equals to 53.43 five. And this is feet per second. So that is a answer for party. Thank you.

Jonathon Brumley · Answer

we have a plane flying into a headwind. We are given its velocity and we&#x27;re told its position at the time of zero is zero. First, we want to figure out its position function. So the position function is the integral off the velocity function. So and I&#x27;ll go ahead and distribute the 30 and that velocity function. So I have 480 minus 30 t square. The end The growth of that would be 480 T minus 10 t cubed, plus a constant. See, we confined because we know the position at zero seconds is zero. So if I fill in zero for s empty and zero for the time, I can see that, see is zero. So if I remove the C value, the position function we have is 480 T minus 10 t cute. Now let&#x27;s look at part B. Second, part of our question wants to know how far the airplane travels in two hours. So we&#x27;re gonna take the position at two hours and subtract from it. Its position at zero hours now, normally, we&#x27;d have to worry about if there was a direction change. But if we look at the original formula. Uh, that&#x27;s going to be a positives longest TIAs at least four. And we just want this for the first two hours. As of two, if I feel that into my position equation, I get 960 minus 80 track tests of zero, but that&#x27;s just zero, So that&#x27;s a total of 880 miles. Finally, the last part is asking us to figure how far the plane has flown when its velocity reaches 400 MPH. Now remember the velocity. If I distribute to 30 the velocity function waas 400 m 80 minus 30 T square. Well, let&#x27;s figure the time that it takes for that plane to reach a velocity of 400 MPH. So we&#x27;ll set that equal to 400 and we&#x27;ll solve that for tea. It&#x27;s attractive. 4 80 away. We have 30 t squared equals negative. 80. Divide by negative. 10 T squared is 8/3 and take the square root, and that gives us a decimal of about 1.63 so it takes about 1.63 hours to get up to a speed of actually down to a speed of 250 Sorry, 400 MPH. So now let&#x27;s figure it&#x27;s position change from zero from a time of 0 to 1.63 hours. That would be when I feel in the 1.63 into our position function. It&#x27;s 480 times 1.63 minus 10 times one point 63 Cute. Make that a little bit leader. 1.63 Cute and we subtract s zero, but zero is zero. That&#x27;s going to give us a final distance of about 700 in 39 miles.

Jacquelyn Trost · Answer

Hey, this problem has three parts. The first part. We&#x27;re going to look at the position function. We want to find what the function is and then graph it. Now, I don&#x27;t have my position function, but I can find a position function. I take my initial position in this case, that is zero, and I add to it the integral of my velocity function from zero to t. Now, just to simplify things, I&#x27;m going to write our velocity function with no parentheses. So that would be 480 minus 30 t squared. And I&#x27;m actually going to write this as an ex d x so that I can substitute my tea in later. Okay, so again, this goes to zero. So my position function is the integral of velocity. In this case, that is 480 x minus 10 x cubed. And I&#x27;m going to evaluate that from zero to t. I plug in my tea. I get 480 t minus 10 t cute. And if I subtract evaluating it, zero, that&#x27;s just zero. So here is my equation. I can simplify that just a little bit. Aiken, Factor out a 10 t and what I have left is 48 minus t squared, and that is my position function now about graphing it. This is a cubic function because I have a t Cubed is my highest degree. So a cubic function will look like one of these two because my T Cube term is negative, it&#x27;s minus t cute minus 10 t cubed. That means the bigger T is, the more negative my function will be. So it&#x27;s going to look like the 1st 1 here. Now I only want to look at it from 0 to 3. Where are my zeros of this function? Well, physician at time T equals zero is zero. So that&#x27;s one of my zeros of the function. The other ones. If I look here if I said that equal to zero and solve T is plus or minus the square root of 48 which is almost seven. So I have a zero at plus and minus seven, which means I&#x27;ve got one way over here. It&#x27;s going to come down. It will come up at zero and come back down again over here at positive seven. I only want to look at this section right there. And if I want to be able to mark on the Y axis where I am right over here, I&#x27;ll make this a little higher. If I plug in T equaling three into my position function, I end up with a value of 1170. So that is a sketch of the position function over that interval that we&#x27;re looking at. Okay, that was part number one. Our second part. We&#x27;re looking at how far something travels. That&#x27;s displacement. So typically to find this place that we look at the position at the end. In this case, that&#x27;s Time T equals two, and I strapped from there. The position at the beginning or time T equals zero again. I know that my position is zero is zero, so I can leave that off. So I&#x27;m really only looking at the position function when tea is, too. If I plug in t equaling two, what I get is 480 times times T r times two, that is 960 minus 10 times two cubed, which is 80. So my position function gives me 880. Okay, The third piece, we want to look at how far it&#x27;s traveled at a velocity. Well, in order to look at our velocity, which go all the way back to that velocity function that we were given. And I want to know at what time that velocity is equal to a given amount. So there&#x27;s our velocity function, so we will set that equal to 400 and I&#x27;m actually going to rewrite that with no parentheses. In it again. 480 minus 30 t squared. Move that over. I get negative. 80 Equalling negative. 30 t squared. Divide by negative 30. And I&#x27;m gonna saw for tea and TV equals approximately 1.633 Okay, so there&#x27;s my teeth. At that time, I hit my desired velocity. So at that time, what is my displacement? How far have I travelled? Well, for that, we&#x27;re going to go to our position function and we&#x27;re going to plug in that time into my position function. And when I do that, I get a position of 740 0.293

Bruce Edelman · Answer

Okay, so Chapter 11 problem. 70 years s. So we have this crap here that shows a acceleration versus time. Um, acceleration meters per second. Square tongue seconds on. This is, uh, obtained very small airplanes traveling at all the straight course. And it says X equals zero at T equals zero and the equal 60 m per second At T equals zero. So part pay says, what&#x27;s the velocity and position at T equals 20 tickets. Okay. Um, so right, Well, we know for the first six seconds, it travels at 60 m per second because we have no acceleration. So for the first six, it doesn&#x27;t go. Um, alright. Goes 60 m per second. That philosophy is it Go anywhere, then acceleration restarts becoming negative, and then positive impact to zero. Now, the total change in exploration is gonna be equal to integrated area in this curve. Uh huh. And if you look, the integrated area is zero. Oh. So the area under that curve for the bottom part top part cancel each other out. So the average acceleration is you know, So if the average acceleration for the whole time is zero, that means the velocity is going to be 60 years per second. Yeah. So now what? It is the, um, position. Well, that&#x27;s 60 m per second for 20 seconds. So X is gonna be 60 times 20 years. 12 100 years. Cool. Um, part B, Part B wants to figure out what is the average velocity during the interval of 6 to 14 seconds. Well, again, the average acceleration. Well, these pretty symmetrical things that came to each other out. Zero you take the integrated or the area under the curve for this whole thing. And divided by the range the area zero, the range of eight seconds. Average acceleration is era. So the average velocity is going to be 60 m per second during that whole range. Suppose that this question is kind of like a trick question. If they asked us from 6 to 10 or 10 to 14 that would have been different questions. But these what

Hariprasad Annamalai · Answer

Harvey one. What we have is the motion of a jet plane that who&#x27;s acceleration with regard to time is described by the Eva acceleration time graph. And we know the initial position is zero and the initial velocity is 300 ft per second. And what we&#x27;re asking solve for is the time at which the jet stops and also construct the velocity versus time in position versus time Crafts. So what we can do is actually take this acceleration versus time graph and Saul for the velocity versus time function just by integrating it, since the acceleration is given to us with respect to time. So just by looking at the acceleration versus time graph, there are three distinct sections that we can use to solve for the velocity versus time. So the initial initial section is from T equals zero to t equals 10 seconds. And as you can see, the acceleration is just zero. Therefore, our velocity with respect that time will simply be our initial velocity. So for tea, so basically four T from zero to 10 seconds or acceleration or velocity with respect to time will just be 300 ft per second. Since there is no acceleration. Next, we&#x27;re gonna look from 10 to 20 seconds. We see that our acceleration is linear. So from 10 to 20 seconds we&#x27;ll see that acceleration is defined by this linear functions. Such that acceleration equals t minus 30 feet per second squared now, using the initial condition that are velocity is 300 ft per second. At 10 seconds, we can say that the integral from 300 to V of D. V is equal to the integral from 10 seconds to T. T minus 30 DT. Now, carrying this interview out on both sides gives us the expression for velocity as a function of time. For this portion from T equals 10 to 20 seconds, that velocity is equal toe one half a T squared minus 30 T plus 550. Next, we can see that from at T equals 20. The acceleration is constant at negative 10 m per second squared. So what we can do is we can say that from 40 greater than 20 seconds that the integral from the velocity at 20 seconds, which we compute to be 150 ft per second. But plugging this in by plugging in 10 seconds or 20 seconds into our equation for velocity that we found. Here we see that the velocity at 20 seconds is 1 50 ft per second. So integrating this from 1 52 p of TV is equal to 22 t of negative 10 DT, which we got from our craft. We see that the velocity as a function of time for tea greater than 20 seconds is negative. 10 t plus 350. Now we can directly use this to solve for the total time that it&#x27;s going to take to bring this aircraft to zero velocity. So what we can simply do is say that zero will equal negative 10 t prime plus 3 50. Solving this equation for T prime gives us that T prime is equal to 35 seconds. So from this, we have to have all the information we need to construct velocity versus time graph, however recalled that were also asked toe plot art position versus time graph. So we don&#x27;t have enough information for that yet. What we need to do is take our velocity functions and integrate them to get our position functions So for our initial time, from 0 to 10 seconds, what we can do is integrate our velocity functions and say that integral from zero s of DS will be equal to then grow from zero to t of 300 DT. This gives us that s is equal to 300 t in feet from 0 to 10 seconds. So by plugging in 10 seconds here, we&#x27;ll see that ad 10 seconds. The position is 3000 ft. What we can do is we can use this piece of information to move to our next portion of acceleration from 10 to 20 seconds when our acceleration is linear. What we can do is say that the integral from 3000 feet toe s of DS is equal to the integral from 10 seconds to t of one half a T squared minus 30 T plus 550 DT computing. This integral gives us that the position from 10 to 20 seconds is equal 2 1/6 it cube minus 15 t squared plus 550 t minus 1167. And by plugging in pretty equal to 20 seconds at 20 seconds. Our position is going to be equal to 5167 ft. Antique was 20. Next. What we can do is for the period from 20 to 35 seconds. Which we computed is the time that it takes for this jet to be brought to rest. We can go ahead and do the integral from 5167 s of DS is equal to the integral from 22 t negative 10 T plus 3 50 DT. This gives us an expression for position that s is equal to negative five t squared plus 350 t plus 1 67. So by plugging in 35 seconds for this, we&#x27;ll see that s at 35 is equal to 6292 ft. Lastly, with this, we have enough information to plot. RS versus T graph is well, so first, what we&#x27;ll do is actually just go ahead and plus our velocity versus time graph. So you have velocity on the Y axis and time on the X axis. So we have velocity and feet per second. And here are the different points that we have. So we have 10 300. So that&#x27;s our non changing velocity. Since we have zero acceleration from 0 to 10 seconds and then a T equals 20 we see a velocity of 1 50 and at equal to 35 we see a velocity equal to survive. So we could go ahead, put these points here and we&#x27;ll see that this is the path that has followed during this course. We have quadratic the quadratic here and win here. Next. What we&#x27;ll do is we can plot the position versus time. So we have a position on the Y axis and time in seconds on the X axis. Position isn&#x27;t feet. So we here are the different times adventurist we have to eat with a 10 20 and 35. And here are different positions. So a T equals 10. We have a position of 3000. At T equals 20. We have a position of 5000, 167 and at equal to 35 we have a position of 6292. So here are these points we could go out and plot those and we can connect them using the acceleration or the velocity that we know. So we know that from 0 to 10 this position traveling simply linear. Then from 10 to 20 we have a third order polynomial. So it&#x27;s cubic and then from 20 to 35 we have a quadratic and this levels off and that&#x27;s our S versus T graph.

Stephen Hobbs · Answer

Based on the graph, we could conjecture that the displacement from 0 to 5 is going to be positive because we have more positive area up here and here combined than we do negative area below the X axis. And so, therefore, that that would be part a conjecture that the displacement is positive. And then, for part B, we could calculate what this is by using a graphing utility. And so we&#x27;re gonna take the integral of this function from 0 to 5 and find out the exact area under the curve. And then it&#x27;s not surprising that it&#x27;s positive the exact value is to point or yeah, by six. And so this is the displacement. And then again, they do not give us any units. And so we just know that this is how many units we are away from the origin, um, or of whatever. Our starting point is after five seconds or, you know, time is five

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Let $\overline{x}=\frac{1}{n}\left(x_{1}+\cdots+x…

Problem 13 Easy Difficulty

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$v(4.5)=53.0 \mathrm{ft} / \mathrm{sec}$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

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Heather Z.

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$v(4.5)=53.0 \mathrm{ft} / \mathrm{sec}$

Linear Algebra and Its Applications

Anna Marie V.

Heather Z.

Kayleah T.

Samuel H.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Anna Marie V.

Heather Z.

Kayleah T.

Samuel H.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications