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[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.

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

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 6

Applications to Linear Models

Vectors

Campbell University

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Harvey Mudd College

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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'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'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'll have 1000 40 equals toe one, We'll have 111 square is 11 Q B is 1 40 equals toe to we'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'll write this as bitter only. And here we'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's say a okay. Times beater. And let'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'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'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's 4.5 is equals to 53.43 five. And this is feet per second. So that is a answer for party. Thank you.

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