Let
and
. Then
can be parameterized by
![\vec r(t)=\cos t\,\vec\imath+\sin t\,\vec\jmath+(\sin^2t-\cos^2t)\,\vec k](https://tex.z-dn.net/?f=%5Cvec%20r%28t%29%3D%5Ccos%20t%5C%2C%5Cvec%5Cimath%2B%5Csin%20t%5C%2C%5Cvec%5Cjmath%2B%28%5Csin%5E2t-%5Ccos%5E2t%29%5C%2C%5Cvec%20k)
with
, and its derivative is
![\dfrac{\mathrm d\vec r}{\mathrm dt}=-\sin t\,\vec\imath+\cos t\,\vec\jmath+4\sin t\cos t\,\vec k](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%5Cvec%20r%7D%7B%5Cmathrm%20dt%7D%3D-%5Csin%20t%5C%2C%5Cvec%5Cimath%2B%5Ccos%20t%5C%2C%5Cvec%5Cjmath%2B4%5Csin%20t%5Ccos%20t%5C%2C%5Cvec%20k)
Now,
![\vec F(x,y,z)=x^2y\,\vec\imath+\dfrac{x^3}3\,\vec\jmath+xy\,\vec k](https://tex.z-dn.net/?f=%5Cvec%20F%28x%2Cy%2Cz%29%3Dx%5E2y%5C%2C%5Cvec%5Cimath%2B%5Cdfrac%7Bx%5E3%7D3%5C%2C%5Cvec%5Cjmath%2Bxy%5C%2C%5Cvec%20k)
![\implies\vec F(\vec r(t))=\cos^2t\sin t\,\vec\imath+\dfrac{\cos^3t}3\,\vec\jmath+\cos t\sin t\,\vec k](https://tex.z-dn.net/?f=%5Cimplies%5Cvec%20F%28%5Cvec%20r%28t%29%29%3D%5Ccos%5E2t%5Csin%20t%5C%2C%5Cvec%5Cimath%2B%5Cdfrac%7B%5Ccos%5E3t%7D3%5C%2C%5Cvec%5Cjmath%2B%5Ccos%20t%5Csin%20t%5C%2C%5Cvec%20k)
Then the work done by
along
is
![\displaystyle\int_C\vec F(x,y,z)\cdot\mathrm d\vec r=\int_0^{2\pi}\vec F(\vec r(t))\cdot\frac{\mathrm d\vec r}{\mathrm dt}\,\mathrm dt=\int_0^{2\pi}\left(3\cos^2t\sin^2t+\frac{\cos^4t}3\right)\,\mathrm dt=\boxed{\pi}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_C%5Cvec%20F%28x%2Cy%2Cz%29%5Ccdot%5Cmathrm%20d%5Cvec%20r%3D%5Cint_0%5E%7B2%5Cpi%7D%5Cvec%20F%28%5Cvec%20r%28t%29%29%5Ccdot%5Cfrac%7B%5Cmathrm%20d%5Cvec%20r%7D%7B%5Cmathrm%20dt%7D%5C%2C%5Cmathrm%20dt%3D%5Cint_0%5E%7B2%5Cpi%7D%5Cleft%283%5Ccos%5E2t%5Csin%5E2t%2B%5Cfrac%7B%5Ccos%5E4t%7D3%5Cright%29%5C%2C%5Cmathrm%20dt%3D%5Cboxed%7B%5Cpi%7D)
Answer:
14 ÷ 10=1.4
Step-by-step explanation:
The expression would be:
3x
x=number of plays
Insert 4:
3(4)
=12
They would 12 yards after 4 plays if losing 3 yards every play.
Answer:
<em>y = -2x - 4</em>
Step-by-step explanation:
<em>y + 1 = -2x - 3</em>
<em>Subtract 1 from both sides to get the y by itself</em>
<em>y = -2x - 3 - 1</em>
<em>Simplify (Combine Like terms)</em>
<em>y = -2x - 4</em>