Answer:
B. Sine.
Step-by-step explanation:
We need to find the value of x which is opposite to the given angle (50 degree). Also we are given the length of the hypotenuse.
Sine = opposite / hypotenuse so we would use the sine.
Its worth committing this mnemonic to memory, which helps to find the required trig ratio:
SOH-CAH-TOA.
SineOpposite/Hypotenuse- CosineAdjacent/Hypotenuse-TangentOpposite/Adjacent.
Answer:
A box plot is drawn with end points at 24 and 49.The box extends from 28 to 44 and a vertical line is drawn inside the box at 34.
Step-by-step explanation:
Ordering the data given :
24,28,32,34,40,44,49
We can calculate the 5 number summary required to give the appropriate boxplot that can be produced :
Minimum = 24
Maximum = 49
Median = 1/2(n+1)th term
n = 7
Median = 1/2(8) = 4th term
Median = 34
Lower quartile, Q1 = 1/4(n+1)th term
n = 7
1/4(8) = 2nd term
Q1 = 28
Upper quartile : 3/4(n+1)th term
n = 7
Q3 = 3/4(8) = 6th term
Q3= 44
The key idea is that, if a vector field is conservative, then it has curl 0. Equivalently, if the curl is not 0, then the field is not conservative. But if we find that the curl is 0, that on its own doesn't mean the field is conservative.
1.
![\mathrm{curl}\vec F=\dfrac{\partial(5x+10y)}{\partial x}-\dfrac{\partial(-6x+5y)}{\partial y}=5-5=0](https://tex.z-dn.net/?f=%5Cmathrm%7Bcurl%7D%5Cvec%20F%3D%5Cdfrac%7B%5Cpartial%285x%2B10y%29%7D%7B%5Cpartial%20x%7D-%5Cdfrac%7B%5Cpartial%28-6x%2B5y%29%7D%7B%5Cpartial%20y%7D%3D5-5%3D0)
We want to find
such that
. This means
![\dfrac{\partial f}{\partial x}=-6x+5y\implies f(x,y)=-3x^2+5xy+g(y)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3D-6x%2B5y%5Cimplies%20f%28x%2Cy%29%3D-3x%5E2%2B5xy%2Bg%28y%29)
![\dfrac{\partial f}{\partial y}=5x+10y=5x+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=10y\implies g(y)=5y^2+C](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D5x%2B10y%3D5x%2B%5Cdfrac%7B%5Cmathrm%20dg%7D%7B%5Cmathrm%20dy%7D%5Cimplies%5Cdfrac%7B%5Cmathrm%20dg%7D%7B%5Cmathrm%20dy%7D%3D10y%5Cimplies%20g%28y%29%3D5y%5E2%2BC)
![\implies\boxed{f(x,y)=-3x^2+5xy+5y^2+C}](https://tex.z-dn.net/?f=%5Cimplies%5Cboxed%7Bf%28x%2Cy%29%3D-3x%5E2%2B5xy%2B5y%5E2%2BC%7D)
so
is conservative.
2.
![\mathrm{curl}\vec F=\left(\dfrac{\partial(-2y)}{\partial z}-\dfrac{\partial(1)}{\partial y}\right)\vec\imath+\left(\dfrac{\partial(-3x)}{\partial z}-\dfrac{\partial(1)}{\partial z}\right)\vec\jmath+\left(\dfrac{\partial(-2y)}{\partial x}-\dfrac{\partial(-3x)}{\partial y}\right)\vec k=\vec0](https://tex.z-dn.net/?f=%5Cmathrm%7Bcurl%7D%5Cvec%20F%3D%5Cleft%28%5Cdfrac%7B%5Cpartial%28-2y%29%7D%7B%5Cpartial%20z%7D-%5Cdfrac%7B%5Cpartial%281%29%7D%7B%5Cpartial%20y%7D%5Cright%29%5Cvec%5Cimath%2B%5Cleft%28%5Cdfrac%7B%5Cpartial%28-3x%29%7D%7B%5Cpartial%20z%7D-%5Cdfrac%7B%5Cpartial%281%29%7D%7B%5Cpartial%20z%7D%5Cright%29%5Cvec%5Cjmath%2B%5Cleft%28%5Cdfrac%7B%5Cpartial%28-2y%29%7D%7B%5Cpartial%20x%7D-%5Cdfrac%7B%5Cpartial%28-3x%29%7D%7B%5Cpartial%20y%7D%5Cright%29%5Cvec%20k%3D%5Cvec0)
Then
![\dfrac{\partial f}{\partial x}=-3x\implies f(x,y,z)=-\dfrac32x^2+g(y,z)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3D-3x%5Cimplies%20f%28x%2Cy%2Cz%29%3D-%5Cdfrac32x%5E2%2Bg%28y%2Cz%29)
![\dfrac{\partial f}{\partial y}=-2y=\dfrac{\partial g}{\partial y}\implies g(y,z)=-y^2+h(y)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D-2y%3D%5Cdfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20y%7D%5Cimplies%20g%28y%2Cz%29%3D-y%5E2%2Bh%28y%29)
![\dfrac{\partial f}{\partial z}=1=\dfrac{\mathrm dh}{\mathrm dz}\implies h(z)=z+C](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20z%7D%3D1%3D%5Cdfrac%7B%5Cmathrm%20dh%7D%7B%5Cmathrm%20dz%7D%5Cimplies%20h%28z%29%3Dz%2BC)
![\implies\boxed{f(x,y,z)=-\dfrac32x^2-y^2+z+C}](https://tex.z-dn.net/?f=%5Cimplies%5Cboxed%7Bf%28x%2Cy%2Cz%29%3D-%5Cdfrac32x%5E2-y%5E2%2Bz%2BC%7D)
so
is conservative.
3.
![\mathrm{curl}\vec F=\dfrac{\partial(10y-3x\cos y)}{\partial x}-\dfrac{\partial(-\sin y)}{\partial y}=-3\cos y+\cos y=-2\cos y\neq0](https://tex.z-dn.net/?f=%5Cmathrm%7Bcurl%7D%5Cvec%20F%3D%5Cdfrac%7B%5Cpartial%2810y-3x%5Ccos%20y%29%7D%7B%5Cpartial%20x%7D-%5Cdfrac%7B%5Cpartial%28-%5Csin%20y%29%7D%7B%5Cpartial%20y%7D%3D-3%5Ccos%20y%2B%5Ccos%20y%3D-2%5Ccos%20y%5Cneq0)
so
is not conservative.
4.
![\mathrm{curl}\vec F=\left(\dfrac{\partial(5y^2)}{\partial z}-\dfrac{\partial(5z^2)}{\partial y}\right)\vec\imath+\left(\dfrac{\partial(-3x^2)}{\partial z}-\dfrac{\partial(5z^2)}{\partial x}\right)\vec\jmath+\left(\dfrac{\partial(5y^2)}{\partial x}-\dfrac{\partial(-3x^2)}{\partial y}\right)\vec k=\vec0](https://tex.z-dn.net/?f=%5Cmathrm%7Bcurl%7D%5Cvec%20F%3D%5Cleft%28%5Cdfrac%7B%5Cpartial%285y%5E2%29%7D%7B%5Cpartial%20z%7D-%5Cdfrac%7B%5Cpartial%285z%5E2%29%7D%7B%5Cpartial%20y%7D%5Cright%29%5Cvec%5Cimath%2B%5Cleft%28%5Cdfrac%7B%5Cpartial%28-3x%5E2%29%7D%7B%5Cpartial%20z%7D-%5Cdfrac%7B%5Cpartial%285z%5E2%29%7D%7B%5Cpartial%20x%7D%5Cright%29%5Cvec%5Cjmath%2B%5Cleft%28%5Cdfrac%7B%5Cpartial%285y%5E2%29%7D%7B%5Cpartial%20x%7D-%5Cdfrac%7B%5Cpartial%28-3x%5E2%29%7D%7B%5Cpartial%20y%7D%5Cright%29%5Cvec%20k%3D%5Cvec0)
Then
![\dfrac{\partial f}{\partial x}=-3x^2\implies f(x,y,z)=-x^3+g(y,z)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3D-3x%5E2%5Cimplies%20f%28x%2Cy%2Cz%29%3D-x%5E3%2Bg%28y%2Cz%29)
![\dfrac{\partial f}{\partial y}=5y^2=\dfrac{\partial g}{\partial y}\implies g(y,z)=\dfrac53y^3+h(z)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D5y%5E2%3D%5Cdfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20y%7D%5Cimplies%20g%28y%2Cz%29%3D%5Cdfrac53y%5E3%2Bh%28z%29)
![\dfrac{\partial f}{\partial z}=5z^2=\dfrac{\mathrm dh}{\mathrm dz}\implies h(z)=\dfrac53z^3+C](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20z%7D%3D5z%5E2%3D%5Cdfrac%7B%5Cmathrm%20dh%7D%7B%5Cmathrm%20dz%7D%5Cimplies%20h%28z%29%3D%5Cdfrac53z%5E3%2BC)
![\implies\boxed{f(x,y,z)=-x^3+\dfrac53y^3+\dfrac53z^3+C}](https://tex.z-dn.net/?f=%5Cimplies%5Cboxed%7Bf%28x%2Cy%2Cz%29%3D-x%5E3%2B%5Cdfrac53y%5E3%2B%5Cdfrac53z%5E3%2BC%7D)
so
is conservative.
Answer:
(0, -1)
Step-by-step explanation:
((4 + (-4))/2, (2 + (-4))/2) =
(0/2, -2/2) =
(0, -1)