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)
(3,-1). You can know the answer by plugging in the x and y values into the equations and it is a solution if both are true.
Using the exponential growth principle, the population of the country in the next 50 years would be 8074764.08722
<u>The</u><u> </u><u>exponential</u><u> </u><u>growth</u><u> </u><u>can</u><u> </u><u>be</u><u> </u><u>modeled</u><u> </u><u>using</u><u> </u><u>the</u><u> </u><u>relation</u><u> </u><u>:</u>
- <em>A</em><em> </em><em>=</em><em> </em><em>initial</em><em> </em><em>population</em><em> </em><em>value</em><em> </em><em>=</em><em> </em><em>3,000,000</em>
- <em>r</em><em> </em><em>=</em><em> </em><em>growth</em><em> </em><em>rate</em><em> </em><em>=</em><em> </em><em>0.02</em><em> </em>
- <em>t</em><em> </em><em>=</em><em> </em><em>time</em><em> </em><em>=</em><em> </em><em>50</em><em> </em><em>years</em><em> </em>
<u>Substituting</u><u> </u><u>the</u><u> </u><u>values</u><u> </u><u>into</u><u> </u><u>the</u><u> </u><u>equation</u><u>,</u><u> </u><u>we'll</u><u> </u><u>have</u><u> </u><u>:</u><u> </u>
= 8074764.08722
Therefore, the population in the next 50 years would be 8074764.08722
Learn more : brainly.com/question/18796573
Answer:
<em>|x - 5.5| = 3.5</em>
Step-by-step explanation:
First, find the number that is equally distant from 2 and 9 on the number line. It is the average of 2 and 9. (2 + 9)/2 = 11/2 = 5.5. How far is the 5.5 from 9 and from 2? It is 3.5 unit away.
|x - 5.5| = 3.5