Answer:
45,90,135,180,225
Step-by-step explanation:
You're trying to find constants
![a_0,a_1,a_2](https://tex.z-dn.net/?f=a_0%2Ca_1%2Ca_2)
such that
![\hat y=a_0+a_1\hat x+a_2{\hat x}^2](https://tex.z-dn.net/?f=%5Chat%20y%3Da_0%2Ba_1%5Chat%20x%2Ba_2%7B%5Chat%20x%7D%5E2)
. Equivalently, you're looking for the least-square solution to the following matrix equation.
![\underbrace{\begin{bmatrix}1&6&6^2\\1&3&3^2\\\vdots&\vdots&\vdots\\1&9&9^2\end{bmatrix}}_{\mathbf A}\underbrace{\begin{bmatrix}a_0\\a_1\\a_2\end{bmatrix}}_{\mathbf x}=\underbrace{\begin{bmatrix}100\\110\\\vdots\\70\end{bmatrix}}_{\mathbf b}](https://tex.z-dn.net/?f=%5Cunderbrace%7B%5Cbegin%7Bbmatrix%7D1%266%266%5E2%5C%5C1%263%263%5E2%5C%5C%5Cvdots%26%5Cvdots%26%5Cvdots%5C%5C1%269%269%5E2%5Cend%7Bbmatrix%7D%7D_%7B%5Cmathbf%20A%7D%5Cunderbrace%7B%5Cbegin%7Bbmatrix%7Da_0%5C%5Ca_1%5C%5Ca_2%5Cend%7Bbmatrix%7D%7D_%7B%5Cmathbf%20x%7D%3D%5Cunderbrace%7B%5Cbegin%7Bbmatrix%7D100%5C%5C110%5C%5C%5Cvdots%5C%5C70%5Cend%7Bbmatrix%7D%7D_%7B%5Cmathbf%20b%7D)
To solve
![\mathbf{Ax}=\mathbf b](https://tex.z-dn.net/?f=%5Cmathbf%7BAx%7D%3D%5Cmathbf%20b)
, multiply both sides by the transpose of
![\mathbf A](https://tex.z-dn.net/?f=%5Cmathbf%20A)
, which introduces an invertible square matrix on the LHS.
![\mathbf{Ax}=\mathbf b\implies\mathbf A^\top\mathbf{Ax}=\mathbf A^\top\mathbf b\implies\mathbf x=(\mathbf A^\top\mathbf A)^{-1}\mathbf A^\top\mathbf b](https://tex.z-dn.net/?f=%5Cmathbf%7BAx%7D%3D%5Cmathbf%20b%5Cimplies%5Cmathbf%20A%5E%5Ctop%5Cmathbf%7BAx%7D%3D%5Cmathbf%20A%5E%5Ctop%5Cmathbf%20b%5Cimplies%5Cmathbf%20x%3D%28%5Cmathbf%20A%5E%5Ctop%5Cmathbf%20A%29%5E%7B-1%7D%5Cmathbf%20A%5E%5Ctop%5Cmathbf%20b)
Computing this, you'd find that
![\mathbf x=\begin{bmatrix}a_0\\a_1\\a_2\end{bmatrix}\approx\begin{bmatrix}121.119\\-3.786\\-0.175\end{bmatrix}](https://tex.z-dn.net/?f=%5Cmathbf%20x%3D%5Cbegin%7Bbmatrix%7Da_0%5C%5Ca_1%5C%5Ca_2%5Cend%7Bbmatrix%7D%5Capprox%5Cbegin%7Bbmatrix%7D121.119%5C%5C-3.786%5C%5C-0.175%5Cend%7Bbmatrix%7D)
which means the first choice is correct.
A + B = 180
A - B = 70
therefore:
A = B + 70
(B + 70) + B = 180 by substitution
2B = 110
B = 55
A = 180 - 55
B is smallest angle at 55 degrees
Answer:
31/12
Step-by-step explanation: