Answer:
B) There is a correlation between start time and amount of tea consumed. There may or may not be causation. Further studies would have to be done to determine this.
Step-by-step explanation:
Correlation in terms of statistics refers to the relationship that exists between two components. The extent to which the two components or data are related to each other is highlighted with correlation. The change in one component brings the same amount of change in the other component. Correlation discloses the way the variables are related.
In the given excerpt, the correlation between start time and amount of tea consumed has been highlighted. This data is required for further research about the cause.
That would be a right triangle.
The differential equation
![M(x,y) \, dx + N(x,y) \, dy = 0](https://tex.z-dn.net/?f=M%28x%2Cy%29%20%5C%2C%20dx%20%2B%20N%28x%2Cy%29%20%5C%2C%20dy%20%3D%200)
is considered exact if
(where subscripts denote partial derivatives). If it is exact, then its general solution is an implicit function
such that
and
.
We have
![M = \tan(x) - \sin(x) \sin(y) \implies M_y = -\sin(x) \cos(y)](https://tex.z-dn.net/?f=M%20%3D%20%5Ctan%28x%29%20-%20%5Csin%28x%29%20%5Csin%28y%29%20%5Cimplies%20M_y%20%3D%20-%5Csin%28x%29%20%5Ccos%28y%29)
![N = \cos(x) \cos(y) \implies N_x = -\sin(x) \cos(y)](https://tex.z-dn.net/?f=N%20%3D%20%5Ccos%28x%29%20%5Ccos%28y%29%20%5Cimplies%20N_x%20%3D%20-%5Csin%28x%29%20%5Ccos%28y%29)
and
, so the equation is indeed exact.
Now, the solution
satisfies
![f_x = \tan(x) - \sin(x) \sin(y)](https://tex.z-dn.net/?f=f_x%20%3D%20%5Ctan%28x%29%20-%20%5Csin%28x%29%20%5Csin%28y%29)
Integrating with respect to
, we get
![\displaystyle \int f_x \, dx = \int (\tan(x) - \sin(x) \sin(y)) \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20f_x%20%5C%2C%20dx%20%3D%20%5Cint%20%28%5Ctan%28x%29%20-%20%5Csin%28x%29%20%5Csin%28y%29%29%20%5C%2C%20dx)
![\implies f(x,y) = -\ln|\cos(x)| + \cos(x) \sin(y) + g(y)](https://tex.z-dn.net/?f=%5Cimplies%20f%28x%2Cy%29%20%3D%20-%5Cln%7C%5Ccos%28x%29%7C%20%2B%20%5Ccos%28x%29%20%5Csin%28y%29%20%2B%20g%28y%29)
and differentiating with respect to
, we get
![f_y = \cos(x) \cos(y) = \cos(x) \cos(y) + \dfrac{dg}{dy}](https://tex.z-dn.net/?f=f_y%20%3D%20%5Ccos%28x%29%20%5Ccos%28y%29%20%3D%20%5Ccos%28x%29%20%5Ccos%28y%29%20%2B%20%5Cdfrac%7Bdg%7D%7Bdy%7D)
![\implies \dfrac{dg}{dy} = 0 \implies g(y) = C](https://tex.z-dn.net/?f=%5Cimplies%20%5Cdfrac%7Bdg%7D%7Bdy%7D%20%3D%200%20%5Cimplies%20g%28y%29%20%3D%20C)
Then the general solution to the exact equation is
![f(x,y) = \boxed{-\ln|\cos(x)| + \cos(x) \sin(y) = C}](https://tex.z-dn.net/?f=f%28x%2Cy%29%20%3D%20%5Cboxed%7B-%5Cln%7C%5Ccos%28x%29%7C%20%2B%20%5Ccos%28x%29%20%5Csin%28y%29%20%3D%20C%7D)
Answer:
<XYZ = 117°
<XYW+ WYZ = 117°
(6x+44)+(-10x+65)= 117°
6x+44-10x+65= 117°
-4x+ 109= 117°
-4x= 117-109
-4x= 8
x= 8/-4
x= -2
<XYW= 6x+44= 6×-2+44= -12+44= 32°
<WYZ= -10x+65 = -10×-2+ 65= 20+65= 85°
Step-by-step explanation: