Answer:
x=4
Step-by-step explanation:
Given M is the midpoint
We know that midpoint of a line divides the line into two equal parts
Therefore on equating we get
![4x + 1 = 8x - 15](https://tex.z-dn.net/?f=4x%20%20%2B%201%20%3D%208x%20-%2015)
![1 + 15 = 8x - 4x](https://tex.z-dn.net/?f=1%20%2B%2015%20%3D%208x%20-%204x)
![16 = 4x](https://tex.z-dn.net/?f=16%20%3D%204x)
![x = 4](https://tex.z-dn.net/?f=x%20%3D%204)
Denote the cylindrical surface by
, and its interior by
. By the divergence theorem, the integral of
across
(the outward flow of the fluid) is equal to the integral of the divergence of
over the space it contains,
:
![\displaystyle\iint_S\vec v\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec v)\,\mathrm dV](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_S%5Cvec%20v%5Ccdot%5Cmathrm%20d%5Cvec%20S%3D%5Ciiint_R%28%5Cnabla%5Ccdot%5Cvec%20v%29%5C%2C%5Cmathrm%20dV)
The given velocity vector has divergence
![\vec v=z\,\vec\imath+y^2\,\vec\jmath+x^2\,\vec k\implies\nabla\cdot\vec v=\dfrac{\partial(z)}{\partial x}+\dfrac{\partial(y^2)}{\partial y}+\dfrac{\partial(x^2)}{\partial z}=2y](https://tex.z-dn.net/?f=%5Cvec%20v%3Dz%5C%2C%5Cvec%5Cimath%2By%5E2%5C%2C%5Cvec%5Cjmath%2Bx%5E2%5C%2C%5Cvec%20k%5Cimplies%5Cnabla%5Ccdot%5Cvec%20v%3D%5Cdfrac%7B%5Cpartial%28z%29%7D%7B%5Cpartial%20x%7D%2B%5Cdfrac%7B%5Cpartial%28y%5E2%29%7D%7B%5Cpartial%20y%7D%2B%5Cdfrac%7B%5Cpartial%28x%5E2%29%7D%7B%5Cpartial%20z%7D%3D2y)
Then the total outward flow is
![\displaystyle\iiint_R2y\,\mathrm dV](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciiint_R2y%5C%2C%5Cmathrm%20dV)
Converting to cylindrical coordinates gives the integral
![\displaystyle2\int_0^{2\pi}\int_0^3\int_0^4r^2\sin\theta\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\boxed{0}](https://tex.z-dn.net/?f=%5Cdisplaystyle2%5Cint_0%5E%7B2%5Cpi%7D%5Cint_0%5E3%5Cint_0%5E4r%5E2%5Csin%5Ctheta%5C%2C%5Cmathrm%20dz%5C%2C%5Cmathrm%20dr%5C%2C%5Cmathrm%20d%5Ctheta%3D%5Cboxed%7B0%7D)
Answer:
The length of board required is 13 feet 6 inches .
Step-by-step explanation:
Board required to make one bird house = 2 feet 3 inches
= 2 x 12 + 3 = 27 inches
Number of bird houses = 6
So, the board required
= 6 x 27 = 162 inches
13.5 feet = 13 feet 6 inches
X = height of bush
12/3.5 = 3.2/x [cross multiply]
(12)(x) = (3.2)(3.5)
12x = 11.2 [Divide 12 on both sides]
x = 0.93 m
The height of the bush is 0.93 meters.
Best of luck.