Answer:
357.769230769
Step-by-step explanation:
nah
![\vec F(x,y,z)=\dfrac74\cos(xyz)\langle yz,xz,xy\rangle](https://tex.z-dn.net/?f=%5Cvec%20F%28x%2Cy%2Cz%29%3D%5Cdfrac74%5Ccos%28xyz%29%5Clangle%20yz%2Cxz%2Cxy%5Crangle)
Computing the line integral directly is cumbersome, if not impossible by elementary means. Let's instead try to determine if
is conservative. We look for a scalar function
such that
. We should have
![\dfrac{\partial f}{\partial x}=yz\cos(xyz)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3Dyz%5Ccos%28xyz%29)
(ignoring the 7/4 for a moment). Integrating both sides wrt
gives
![\displaystyle\int\cos(xyz)yz\,\mathrm dx=\sin(xyz)+g(y,z)](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Ccos%28xyz%29yz%5C%2C%5Cmathrm%20dx%3D%5Csin%28xyz%29%2Bg%28y%2Cz%29)
Then differentiating wrt
gives
![\dfrac{\partial(\sin(xyz))}{\partial y}=xz\cos(xyz)=xz\cos(xyz)+\dfrac{\partial g}{\partial y}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%28%5Csin%28xyz%29%29%7D%7B%5Cpartial%20y%7D%3Dxz%5Ccos%28xyz%29%3Dxz%5Ccos%28xyz%29%2B%5Cdfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20y%7D)
![\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)](https://tex.z-dn.net/?f=%5Cimplies%5Cdfrac%7B%5Cpartial%20g%7D%7B%5Cpartial%20y%7D%3D0%5Cimplies%20g%28y%2Cz%29%3Dh%28z%29)
Differentiating wrt
gives
![\dfrac{\partial(\sin(xyz))}{\partial z}=xy\cos(xyz)=xy\cos(xyz)+\dfrac{\mathrm dh}{\mathrm dz}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%28%5Csin%28xyz%29%29%7D%7B%5Cpartial%20z%7D%3Dxy%5Ccos%28xyz%29%3Dxy%5Ccos%28xyz%29%2B%5Cdfrac%7B%5Cmathrm%20dh%7D%7B%5Cmathrm%20dz%7D)
![\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C](https://tex.z-dn.net/?f=%5Cimplies%5Cdfrac%7B%5Cmathrm%20dh%7D%7B%5Cmathrm%20dz%7D%3D0%5Cimplies%20h%28z%29%3DC)
So we have (and here we re-introduce the 7/4)
![f(x,y,z)=\dfrac74\sin(xyz)+C](https://tex.z-dn.net/?f=f%28x%2Cy%2Cz%29%3D%5Cdfrac74%5Csin%28xyz%29%2BC)
and by the fundamental theorem of calculus,
![\displaystyle\int_C\nabla f\cdot\mathrm d\vec r=f(\vec b)-f(\vec a)](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_C%5Cnabla%20f%5Ccdot%5Cmathrm%20d%5Cvec%20r%3Df%28%5Cvec%20b%29-f%28%5Cvec%20a%29)
where
and
are vectors representing the start- and endpoints of
. So
![\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\frac74\sin\frac\pi6-\frac74\sin\frac\pi2=\boxed{\frac78}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_C%5Cvec%20F%5Ccdot%5Cmathrm%20d%5Cvec%20r%3D%5Cfrac74%5Csin%5Cfrac%5Cpi6-%5Cfrac74%5Csin%5Cfrac%5Cpi2%3D%5Cboxed%7B%5Cfrac78%7D)
Answer: ![(gof)(-7) =495](https://tex.z-dn.net/?f=%28gof%29%28-7%29%20%3D495)
Step-by-step explanation:
Given the functions f(x) and g(x), to find
you need to substitute
into the function g(x):
![(gof)(x) = ( x^2 + 6)+ 8( x^2 + 6)\\\\(gof)(x)=x^2 + 6+ 8x^2 + 48\\\\(gof)(x)=9x^2 + 54](https://tex.z-dn.net/?f=%28gof%29%28x%29%20%3D%20%28%20x%5E2%20%2B%206%29%2B%208%28%20x%5E2%20%2B%206%29%5C%5C%5C%5C%28gof%29%28x%29%3Dx%5E2%20%2B%206%2B%208x%5E2%20%2B%2048%5C%5C%5C%5C%28gof%29%28x%29%3D9x%5E2%20%2B%2054)
Now, to find
you must substitute
into
, then you get:
![(gof)(-7) = 9(-7)^2 + 54\\\\(gof)(-7) =9(49)+54\\\\(gof)(-7) =441+54\\\\(gof)(-7) =495](https://tex.z-dn.net/?f=%28gof%29%28-7%29%20%3D%209%28-7%29%5E2%20%2B%2054%5C%5C%5C%5C%28gof%29%28-7%29%20%3D9%2849%29%2B54%5C%5C%5C%5C%28gof%29%28-7%29%20%3D441%2B54%5C%5C%5C%5C%28gof%29%28-7%29%20%3D495)
It is, the FitnessGram™ Pacer Test is a multistage aerobic capacity test that progressively gets more difficult as it continues. The 20 meter pacer test will begin in 30 seconds. ... The running speed starts slowly, but gets faster each minute after you hear this signal
Answer:
the rate of precession is 0.041 rad/s
Step-by-step explanation:
given information:
radius, R = 50 cm= 0.5 m
disk length, d = 11 cm, r = 5.5 cm = 0.055
angle, θ = 30°
angular velocity of the disk, ω = 1000 rev/min = 1000 2π/60 =
rate precession, Ω
Ω = m g r / I ω
where
m = mass
g = gravitational force
r = the radius of the center disk
I = inertia
ω = angular veocity
I = ![\frac{1}{2} mR^{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20mR%5E%7B2%7D)
Ω = (m g r )/ (
ω)
= 2 g r / (
ω)
= 2 x 9.8 x 0.055 / (
104.7)
= 0.041 rad/s