Answer:
The answer would be 27π
Step-by-step explanation:
the area is 36pi and the shaded region is 3/4 of the circle, as a 90 degree angle is 1/4 of a 360 degree circle. 3/4 of 36pi is 27pi
![\dfrac{\mathrm dy}{\mathrm dx}=\cos(x+y)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%3D%5Ccos%28x%2By%29)
Let
, so that
:
![\dfrac{\mathrm dv}{\mathrm dx}=\cos v+1](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dv%7D%7B%5Cmathrm%20dx%7D%3D%5Ccos%20v%2B1)
Now the ODE is separable, and we have
![\dfrac{\mathrm dv}{1+\cos v}=\mathrm dx](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dv%7D%7B1%2B%5Ccos%20v%7D%3D%5Cmathrm%20dx)
Integrating both sides gives
![\displaystyle\int\frac{\mathrm dv}{1+\cos v}=\int\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20dv%7D%7B1%2B%5Ccos%20v%7D%3D%5Cint%5Cmathrm%20dx)
For the integral on the left, rewrite the integrand as
![\dfrac1{1+\cos v}\cdot\dfrac{1-\cos v}{1-\cos v}=\dfrac{1-\cos v}{1-\cos^2v}=\csc^2v-\csc v\cot v](https://tex.z-dn.net/?f=%5Cdfrac1%7B1%2B%5Ccos%20v%7D%5Ccdot%5Cdfrac%7B1-%5Ccos%20v%7D%7B1-%5Ccos%20v%7D%3D%5Cdfrac%7B1-%5Ccos%20v%7D%7B1-%5Ccos%5E2v%7D%3D%5Ccsc%5E2v-%5Ccsc%20v%5Ccot%20v)
Then
![\displaystyle\int\frac{\mathrm dv}{1+\cos v}=-\cot v+\csc v+C](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20dv%7D%7B1%2B%5Ccos%20v%7D%3D-%5Ccot%20v%2B%5Ccsc%20v%2BC)
and so
![\csc v-\cot v=x+C](https://tex.z-dn.net/?f=%5Ccsc%20v-%5Ccot%20v%3Dx%2BC)
![\csc(x+y)-\cot(x+y)=x+C](https://tex.z-dn.net/?f=%5Ccsc%28x%2By%29-%5Ccot%28x%2By%29%3Dx%2BC)
Given that
, we find
![\csc\left(0+\dfrac\pi2\right)-\cot\left(0+\dfrac\pi2\right)=0+C\implies C=1](https://tex.z-dn.net/?f=%5Ccsc%5Cleft%280%2B%5Cdfrac%5Cpi2%5Cright%29-%5Ccot%5Cleft%280%2B%5Cdfrac%5Cpi2%5Cright%29%3D0%2BC%5Cimplies%20C%3D1)
so that the particular solution to this IVP is
![\csc(x+y)-\cot(x+y)=x+1](https://tex.z-dn.net/?f=%5Ccsc%28x%2By%29-%5Ccot%28x%2By%29%3Dx%2B1)
Answer:
the distance is -2
Step-by-step explanation:
because you subtract
They would go around the multiplication so it should look like 7+(9 x 3) -1=25
Answer:
5/15
Step-by-step explanation:
there are 15 balls altogether, 5 of which are blue meaning there is a 5/15 chance of picking a blue ball which as a percentage is 33.33% chance